Flexibility for the Three-Dimensional Navier–Stokes Equations via Moving Hill Vortices

Quoc-Hung Nguyen Thanks: State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; Institute of Mathematics, Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, Beijing 100190, China. Email: qhnguyen@amss.ac.cn. Zexi Wang Thanks: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, PR China. Email: wangzexi25@mails.ucas.ac.cn.

Abstract

We construct weak solutions of the three-dimensional incompressible Navier–Stokes equations on the torus. The convex-integration scheme is based on the moving-dipole construction of Bruè, Colombo, and Kumar [14]. For the explicit exponent

\bar{p}=\frac{6}{5}+5\times 10^{-5},

and for any two mean-zero, divergence-free vector fields in $L^{2}(\mathbb{T}^{3})$ , we construct a weak solution whose traces at times $0$ and $1$ approximate the prescribed fields arbitrarily well and which satisfies

u\in C([0,1];L^{2}(\mathbb{T}^{3})),\qquad\nabla u\in C([0,1];L^{\bar{p}}(\mathbb{T}^{3})).

Exploiting the time-locality of the iteration, we also obtain exact nonuniqueness for a dense set of initial data in $L^{2}_{\sigma}(\mathbb{T}^{3})$ . The principal perturbations are localized, rescaled copies of Hill’s spherical vortex. The Hill scaling preserves both the kinetic-energy scale and the $L^{6/5}$ -scale of the velocity gradient. The construction uses localization of the potential exterior flow, long-orbit averaging of moving vortex cores, an auxiliary source correction, and a temporal corrector compatible with uniform-in-time Sobolev control.

1 Introduction

1.1 The problem and the main result

We consider the incompressible Navier–Stokes equations on the three-dimensional torus:

\begin{cases}\partial_{t}u+\operatorname{div}(u\otimes u)-\nu\Delta u+\nabla p=0,&(x,t)\in\mathbb{T}^{3}\times(0,1),\\ \operatorname{div}u=0,\end{cases}

(1.1)

where $u:\mathbb{T}^{3}\times[0,1]\to\mathbb{R}^{3}$ is the velocity, $p$ is the pressure, and $\nu>0$ is fixed. We take $\mathbb{T}^{3}=(\mathbb{R}/\mathbb{Z})^{3}$ with normalized Lebesgue measure $|\mathbb{T}^{3}|=1$ . Throughout the paper, all velocity fields have zero spatial mean.

We write

L^{2}_{\sigma}(\mathbb{T}^{3}):=\left\{v\in L^{2}(\mathbb{T}^{3};\mathbb{R}^{3}):\operatorname{div}v=0,\quad\int_{\mathbb{T}^{3}}v(x)\,\mathrm{d}x=0\right\}.

A finite-energy weak solution of (1.1) is a vector field

u\in C([0,1];L^{2}_{\sigma}(\mathbb{T}^{3}))

such that

\displaystyle\int_{0}^{1}\!\!\int_{\mathbb{T}^{3}}\left[u\cdot\partial_{t}\varphi+(u\otimes u):\nabla\varphi+\nu u\cdot\Delta\varphi\right]\,\mathrm{d}x\,\mathrm{d}t+\int_{\mathbb{T}^{3}}u(x,0)\cdot\varphi(x,0)\,\mathrm{d}x=0

(1.2)

for every $\varphi\in C_{c}^{\infty}(\mathbb{T}^{3}\times[0,1);\mathbb{R}^{3})$ satisfying $\operatorname{div}\varphi=0$ . The pressure can then be recovered as a distribution. No energy inequality is included in this definition.

Our main result is the following.

Theorem 1.1 (Flexibility).

Let

\bar{p}=\frac{6}{5}+5\times 10^{-5}.

For every $\varepsilon>0$ and every $u^{(0)},u^{(1)}\in L^{2}_{\sigma}(\mathbb{T}^{3})$ , there exists a weak solution $(u,p)$ of (1.1), in the sense of (1.2), such that

u\in C([0,1];L^{2}(\mathbb{T}^{3})),\qquad\nabla u\in C([0,1];L^{\bar{p}}(\mathbb{T}^{3})).

Moreover,

\|u(\cdot,0)-u^{(0)}\|_{L^{2}(\mathbb{T}^{3})}<\varepsilon,\qquad\|u(\cdot,1)-u^{(1)}\|_{L^{2}(\mathbb{T}^{3})}<\varepsilon.

The numerical value of $\bar{p}$ is not claimed to be optimal. It is an explicit exponent for which all inequalities in the parameter selection are strict. The threshold $6/5$ comes from the Hill scaling used in the present construction. Also, Theorem 1.1 is not a statement in the Leray–Hopf class: the regularity $\nabla u\in C_{t}L_{x}^{\bar{p}}$ , with $\bar{p}$ close to $6/5$ , does not imply $u\in L_{t}^{2}H_{x}^{1}$ , and the solutions constructed here are not asserted to satisfy the energy inequality.

1.2 Interpretation and consequences of the theorem

Theorem 1.1 can be restated as a density result for pairs of endpoint traces. Let $\mathcal{S}_{\bar{p}}$ denote the set of all weak solutions of (1.1) satisfying $u\in C([0,1];L^{2}_{\sigma}(\mathbb{T}^{3}))$ and $\nabla u\in C([0,1];L^{\bar{p}}(\mathbb{T}^{3}))$ .

Then

\overline{\left\{\bigl(u(\cdot,0),u(\cdot,1)\bigr):u\in\mathcal{S}_{\bar{p}}\right\}}^{\,L^{2}\times L^{2}}=L^{2}_{\sigma}(\mathbb{T}^{3})\times L^{2}_{\sigma}(\mathbb{T}^{3}).

(1.3)

Thus weak trajectories in this class connect arbitrarily small neighborhoods of any two prescribed states. No compatibility condition is imposed on the endpoint targets.

This is an approximate, rather than exact, controllability statement. Theorem 1.1 does not assert that both prescribed traces are attained exactly. The density relation (1.3) states only that the set of endpoint pairs of weak solutions is dense in the product energy topology. Nevertheless, the construction contains additional information that is not visible in this density statement: the iteration is local in time. Starting from two Reynolds flows that agree near $t=0$ , the successive perturbations can be chosen so that their limits continue to agree on a nontrivial initial time interval. Choosing two distinct terminal targets then gives the following exact same-data consequence.

Theorem 1.2 (Nonuniqueness).

There exists a dense set $\mathcal{D}\subset L^{2}_{\sigma}(\mathbb{T}^{3})$ such that every $u_{\rm in}\in\mathcal{D}$ is the initial datum of at least two distinct weak solutions $u^{(1)},u^{(2)}\in\mathcal{S}_{\bar{p}}$ . More precisely, the two solutions may be chosen so that

u^{(1)}(\cdot,t)=u^{(2)}(\cdot,t)\quad\text{for every }t\in[0,t_{*}]

for some $t_{*}>0$ , while $u^{(1)}(\cdot,1)\neq u^{(2)}(\cdot,1)$ .

Thus the same scheme yields exact nonuniqueness on a dense subset of the finite-energy data. The theorem does not claim nonuniqueness for every prescribed element of $L^{2}_{\sigma}(\mathbb{T}^{3})$ , and the resulting solutions are not asserted to satisfy the energy inequality.

The flexibility is incompatible with the Leray–Hopf energy inequality for suitable target pairs. Indeed, take $u^{(0)}=0$ , $u^{(1)}\neq 0$ , and $0<\varepsilon<\|u^{(1)}\|_{L^{2}}/3$ . The solution supplied by the theorem satisfies

\|u(\cdot,0)\|_{L^{2}}<\varepsilon,\qquad\|u(\cdot,1)\|_{L^{2}}>\|u^{(1)}\|_{L^{2}}-\varepsilon>2\varepsilon,

and therefore cannot obey a nonincreasing kinetic-energy inequality. Hence the solution does not belong to the Leray–Hopf class.

The theorem gives the following small improvement over $L_{x}^{2}$ . Since

\bar{p}=\frac{24001}{20000},\qquad\bar{p}^{\ast}:=\frac{3\bar{p}}{3-\bar{p}}=\frac{72003}{35999}=2+\frac{5}{35999},

Poincaré’s inequality and Sobolev embedding give

u\in C([0,1];W^{1,\bar{p}}(\mathbb{T}^{3}))\hookrightarrow C([0,1];L^{\bar{p}^{\ast}}(\mathbb{T}^{3})),\qquad\bar{p}^{\ast}>2.

This is a fixed-time spatial gain, rather than an estimate obtained by integrating a stronger norm in time. The class nevertheless remains far below the classical uniqueness range.

1.3 Background and related work

Leray’s classical construction [76] on $\mathbb{R}^{3}$ , and Hopf’s construction [61] in the periodic and bounded-domain settings, give global weak solutions in

u\in L_{t}^{\infty}L_{x}^{2}\cap L_{t}^{2}\dot{H}_{x}^{1}

satisfying the energy inequality. Uniqueness holds under the Ladyzhenskaya–Prodi–Serrin conditions

u\in L_{t}^{s}L_{x}^{q},\qquad\frac{2}{s}+\frac{3}{q}\leq 1,\qquad q>3,

with the usual endpoint refinements; see [88, 94, 72], and [46] for the endpoint $q=3$ . Quantitative refinements are discussed in [97]. The global regularity problem for smooth finite-energy data remains open [47]. For an introduction to the mathematical analysis of the Euler and Navier–Stokes equations, including the classical well-posedness theory and the Leray–Hopf framework, we refer to the book of Bedrossian and Vicol [8]. Further treatments of weak, mild, and local-energy solutions can be found in [74, 75, 99]. Our result concerns flexibility in a distributional class with finite kinetic energy and a uniform spatial-gradient bound below the energy-dissipation level; the energy inequality is not imposed.

For comparison, we recall the Cauchy theory in critical spaces. Kato [69] proved local well-posedness in $L^{3}(\mathbb{R}^{3})$ and global existence for small data, while Koch and Tataru [70] extended the small-data theory to $BMO^{-1}$ . Bourgain and Pavlović [10] proved norm inflation in $\dot{B}^{-1,\infty}_{\infty}$ , and Germain [52] studied the associated second-iterate instability. Jia and Šverák constructed forward self-similar solutions from large $(-1)$ -homogeneous data and formulated spectral conditions for same-data nonuniqueness [67, 68]; Guillod and Šverák [59] investigated this bifurcation scenario numerically.

Convex integration originates in the geometric flexibility of Nash and Kuiper [83, 71] and Gromov [57, 58]. For incompressible Euler, the modern theory developed from the early constructions of Scheffer and Shnirelman [93, 95, 96] and the differential-inclusion framework of De Lellis and Székelyhidi [40, 42]; see also [39]. After a series of partial results, including [20, 22], Isett [64] proved the flexible part of Onsager’s conjecture [86] by constructing non-energy-conserving Euler solutions in $C_{t}C_{x}^{\alpha}$ for every $\alpha<1/3$ . Together with the energy-conservation result of Constantin, E, and Titi [37] above the exponent $1/3$ , this resolved Onsager’s conjecture. Buckmaster, De Lellis, Székelyhidi, and Vicol [23] subsequently gave a considerably shorter proof of the flexible part. They also constructed solutions with any prescribed smooth positive energy profile and established an $h$ -principle in $C^{\beta}_{t,x}$ for every $\beta<1/3$ . Intermittent and localized variants were developed in [24, 85, 53, 54]; related developments include the two-dimensional Newton–Nash scheme [56] and the construction with prescribed generalized helicity [55]. For surveys, see [98, 28]. The relation between convex integration, intermittency, and phenomenological models of turbulence is discussed by Buckmaster and Vicol in [27]. At the level of the Euler–Reynolds system, these schemes successively replace a coarse stress by the quadratic self-interaction of a rapidly oscillating perturbation. Intermittency adds spatial concentration to this mechanism and gives estimates in Lebesgue spaces below $L^{2}$ , which are used to control the viscous error.

For the viscous problem, spatial concentration makes high-frequency errors small in spaces below $L^{2}$ . Buckmaster and Vicol [26] constructed finite-energy weak solutions on $\mathbb{T}^{3}$ with arbitrary prescribed smooth nonnegative energy profiles. For some $\beta>0$ , these solutions satisfy

v\in C\bigl([0,T];H^{\beta}(\mathbb{T}^{3})\cap W^{1,1+\beta}(\mathbb{T}^{3})\bigr).

In particular, their construction implies nonuniqueness in the class of finite-energy weak solutions. Their building blocks are intermittent Beltrami waves: stationary high-frequency flows modulated by concentrated Dirichlet kernels. The resulting sub- $L^{2}$ estimates make both the Reynolds and viscous errors perturbative, while introducing several separated spatial and temporal scales.

Buckmaster, Colombo, and Vicol [21] combined convex integration with gluing. Starting from two strong Navier–Stokes solutions, they produced a weak solution agreeing with the first near the initial time and with the second near the terminal time, with the same type of uniform-in-time regularity and smoothness outside a fractal set of singular times. Their result also concerns time-uniform regularity. In our setting, however, the endpoint states are arbitrary finite-energy targets and are approximated rather than joined through prescribed strong trajectories. For comparison, Leray–Hopf solutions have a singular time set of Hausdorff dimension at most $1/2$ [76, 92], while the space–time partial-regularity theorem of Caffarelli, Kohn, and Nirenberg [29] controls the parabolic Hausdorff measure of the singular set for suitable weak solutions. The solutions in [21] are not suitable weak solutions, so these partial-regularity results do not apply directly. Buckmaster, Colombo, and Vicol instead proved that the singular set of times has Hausdorff dimension strictly less than (1).

Cheskidov and Luo [31] proved nonuniqueness in $L_{t}^{s}L_{x}^{\infty}$ for every $1\leq s<2$ . More precisely, for every $1\leq s<2$ and $1\leq q<\infty$ , they constructed non-Leray–Hopf weak solutions in

L_{t}^{s}L_{x}^{\infty}\cap L_{t}^{1}W_{x}^{1,q}.

Their construction combines gluing, temporal concentration, spatially intermittent Mikado flows, and a temporally oscillating corrector. The required smallness is obtained in time-integrated norms. Theorem 1.1, by contrast, requires estimates that are uniform at each time, so the principal perturbations must also provide sufficient spatial concentration.

In two dimensions, the same authors proved an $L^{2}$ -critical nonuniqueness result [32]. Their solutions are uniformly continuous in $L^{p}$ for every $p<2$ . This conclusion is uniform in time, as is ours, but their construction uses space–time intermittent accelerating jets, whereas the present construction uses moving Hill vortices.

Related results concern Navier–Stokes equations with fractional dissipation. Luo and Titi [77] proved nonuniqueness for the hyperviscous equation with dissipation $(-\Delta)^{\alpha}$ whenever $\alpha<5/4$ , showing the sharpness of the J.-L. Lions exponent. Nonuniqueness results in the hypodissipative regime were obtained in [35, 43]. In these works the dissipation exponent is varied. Here the standard Laplacian is fixed, and the aim is to obtain time-uniform Sobolev regularity while retaining enough spatial concentration to control the viscous error.

An inviscid result with a related endpoint statement is due to Novack and Vicol [85]. For every $0<\beta<1/2$ , their Euler solutions connect arbitrary divergence-free $L^{2}$ states up to a small endpoint error while remaining in

C_{t}\left(H_{x}^{\beta}\cap L_{x}^{\frac{2-2\beta}{1-2\beta}}\right).

Their construction uses a scheme with higher-order Reynolds stresses, which are corrected through a combinatorial placement of intermittent pipe flows with optimal relative intermittency. Although the endpoint statement is close to Theorem 1.1, their equation is inviscid. In the present setting, the Laplacian of a concentrated perturbation creates an additional principal constraint and requires a different choice of scales.

A distinct route to nonuniqueness is based on spectral instability. Albritton, Bruè, and Colombo constructed two suitable Leray–Hopf solutions on $\mathbb{R}^{3}$ with the same force and zero initial data [2]. Their argument starts from an unstable self-similar vortex-ring profile and constructs a second trajectory on its unstable manifold. Thus it gives exact nonuniqueness for a forced Cauchy problem, rather than endpoint flexibility for the unforced equation. An inner–outer gluing argument later extended the construction to the torus and smooth bounded domains [3]. This instability mechanism is based on Vishik’s construction for the forced two-dimensional Euler equations [100, 101]; see also [4] for a detailed presentation. In a different direction, Bruè and De Lellis [16] constructed forced classical Navier–Stokes solutions exhibiting anomalous dissipation in the vanishing-viscosity limit. For a different mechanism producing anomalous diffusion through fractal homogenization, see Armstrong and Vicol [7].

At critical regularity, Coiculescu and Palasek [34] proved exact nonuniqueness for the unforced equation from a particular datum in $BMO^{-1}(\mathbb{T}^{3})\setminus L^{2}(\mathbb{T}^{3})$ . Their nested-Mikado cascade builds on the dyadic scenario of [87] and is not a convex-integration scheme. It branches from one particular infinite-energy datum and shows that smallness is essential in the Koch–Tataru theory; this is complementary to the finite-energy endpoint flexibility obtained here. A recent preprint by Hou, Wang, and Yang [62] presents what the authors describe as a computer-assisted proof of nonuniqueness for Leray–Hopf solutions of the unforced three-dimensional Navier–Stokes equations on $\mathbb{R}^{3}$ . This result is not used in the present paper. Ionescu, Jia, and Palasek [63] obtain a conditional result based on a candidate axisymmetric profile whose required existence remains open. For an overview of instability and convex integration, see Colombo [36]. These results concern three different forms of nonuniqueness. Instability arguments branch from a specially constructed coherent profile; critical-space cascades branch from a particular rough datum; convex integration prescribes a sequence of stresses at each stage. The resulting convex-integration solutions generally lie below the Leray–Hopf class. Theorem 1.1 should be read in this sense: it gives density of attainable endpoint traces in the finite-energy space, while the nonuniqueness statement deduced later concerns a dense set of initial data, not every datum and not the Leray–Hopf class.

The two-dimensional Euler equation provides relevant background for the moving-profile construction. Yudovich’s theorem [103] gives global well-posedness for bounded vorticity, while uniqueness for unforced solutions with vorticity merely in $L^{p}$ , $p<\infty$ , remains open. At the borderline of integrability, Bruè and Colombo [12] first constructed nonunique weak Euler solutions with uniformly bounded kinetic energy and vorticity in the Lorentz space $L^{1,\infty}$ . Buck and Modena [18, 19] subsequently extended this line to the real Hardy spaces $H^{p}$ , first for $2/3<p<1$ , and later, with compact spatial support, for the full range $0<p<1$ . Related results for transport equations appear in [81, 82, 80, 13, 15]; their sharpness is measured against the renormalization and well-posedness theories of DiPerna and Lions [44] and Ambrosio [5].

The present construction is based most directly on the work of Bruè, Colombo, and Kumar [14]. For

1<p<1+\frac{1}{6500},

they construct a dense set of initial data in $L^{2}(\mathbb{T}^{2})\cap W^{1,p}(\mathbb{T}^{2})$ giving rise to infinitely many nonconservative weak Euler solutions in $C_{t}L_{x}^{2}$ with vorticity in $C_{t}L_{x}^{p}$ . Their principal perturbation is a Lamb–Chaplygin dipole concentrated in a moving ball; see [73, §165] and [79] for classical accounts of this structure. The Reynolds stress is decomposed into rank-one directions, with only one direction activated on each short time interval. The radius and speed of the dipole are coupled to the coefficient of that stress component, while its motion along a long rational trajectory converts a localized source into the directional flux required for cancellation. This dynamical use of a traveling coherent structure is also used in the present paper. The passage from that construction to three-dimensional Navier–Stokes is not formal. A planar dipole carries scalar vorticity and travels in a preferred direction, whereas Hill’s vortex is an axisymmetric three-dimensional velocity field whose gradient must be estimated in all directions. Moreover, rescaling the core simultaneously changes its energy, its gradient, its transport speed, and the size of the viscous defect. Balancing these effects is what produces the exponent near $6/5$ and distinguishes the present iteration from its Euler precursor.

For SQG, the weak existence theory was developed in [90, 78]. Buckmaster, Shkoller, and Vicol [25] constructed the first nonunique weak solutions of the unforced equation, and Isett and Ma [66] later gave a direct scalar-level construction. In related work, Bruè, Jin, and Nguyen [17] develop a moving-profile convex-integration scheme for the inviscid SQG equation on $\mathbb{T}^{2}$ . For the explicit, nonoptimized exponent $\bar{p}=4/3+10^{-5}$ , their construction approximately connects any two prescribed mean-zero states in $L^{\bar{p}}(\mathbb{T}^{2})$ by a momentum weak solution in $C([0,1];L^{\bar{p}}(\mathbb{T}^{2}))$ . It also gives nonuniqueness for a dense subset of the mean-zero space $L^{\bar{p}}(\mathbb{T}^{2})$ . The scheme uses localized traveling SQG profiles whose centers move along rational directions, and a bilinear null-form estimate controls the derivative loss caused by the nonlocal relation $u=\Lambda v$ .

The present paper follows the moving-profile strategy introduced by Bruè, Colombo, and Kumar [14] for the two-dimensional Euler equations. In passing to the three-dimensional viscous setting, the Lamb–Chaplygin dipoles used there are replaced by Hill’s spherical vortex, control of the scalar vorticity is replaced by control of the full velocity gradient, and viscosity introduces an additional error at every stage of the iteration. The related SQG work is a separate adaptation of the same general strategy and is not used in the proof of the present paper.

1.4 Moving Hill vortices and the critical scaling

Our construction uses a dynamical cancellation. Hill’s vortex is an exact traveling Euler flow, so transport and self-interaction cancel before antidivergence. After localization, motion along a rational orbit converts the remaining source into a directional stress after time averaging. The Hill scaling preserves the $L^{2}$ -norm and the $L^{6/5}$ -gradient norm, and disjoint time scheduling removes interactions between principal directions.

Figures 1–3 illustrate this three-dimensional version of the mechanism in [14].

Figure 1: One time cell is divided into nine subintervals with disjoint cutoffs. Thus at most one block, associated with

\xi_{i}

, is active at a given time. The arrows are schematic projections of the nine directions in

\mathbb{T}^{3}

The Hill profile.

Hill’s spherical vortex [60, 73] is an axisymmetric, no-swirl traveling Euler flow whose azimuthal vorticity is supported in a ball. Its orbital and linear stability under axisymmetric perturbations were studied in [33, 89]; these results are not used below.

After rotation and normalization, the profile $H_{e}$ satisfies

(e\cdot\nabla)H_{e}+\operatorname{div}(H_{e}\otimes H_{e})+\nabla P_{e}=0,\qquad\operatorname{div}H_{e}=0.

Its exterior potential-doublet form permits localization without changing the core; see Figure 2.

Figure 2: Meridional schematic of the Hill building block. The potential doublet in panel (a) is singular at the origin. Hill’s spherical vortex in panel (b) replaces this singularity by a spherical vorticity core. Outside the unit sphere, the velocity is unchanged and equals

H=\nabla(z/(2r^{3}))

. Rotation about the symmetry axis gives an azimuthal vorticity field supported in the core, and the vortex translates in the direction

-e

For $R>0$ , define

H_{R,e}(x):=R^{-1}H_{e}\!\left(\frac{x}{R^{2/3}}\right),\qquad P_{R,e}(x):=R^{-2}P_{e}\!\left(\frac{x}{R^{2/3}}\right).

The core radius is $R^{2/3}$ , the velocity amplitude is $R^{-1}$ , and

\|H_{R,e}\|_{L^{q}}\lesssim_{q}R^{\frac{2}{q}-1},\qquad\|\nabla H_{R,e}\|_{L^{p}}\lesssim_{p}R^{\frac{2}{p}-\frac{5}{3}}.

Here $1<q\leq\infty$ and $1\leq p\leq\infty$ . Hence the energy and the $L^{6/5}$ -gradient norm are invariant, in agreement with $W^{1,6/5}(\mathbb{T}^{3})\hookrightarrow L^{2}(\mathbb{T}^{3})$ . For $p>6/5$ , however,

\|\nabla H_{R,e}\|_{L^{p}}=R^{-\theta(p)}\|\nabla H_{e}\|_{L^{p}},\qquad\theta(p):=\frac{5}{3}-\frac{2}{p}>0,

so concentration creates a loss that must be offset by the amplitude and averaging parameters.

This Hill threshold differs from the Navier–Stokes scaling threshold. Under

u(x,t)\mapsto u_{\lambda}(x,t):=\lambda u(\lambda x,\lambda^{2}t),

one has

\|\nabla u_{\lambda}\|_{L^{\infty}(\lambda^{-2}I;L_{x}^{p})}=\lambda^{2-3/p}\|\nabla u\|_{L^{\infty}(I;L_{x}^{p})}.

Thus the scale-invariant exponent is $3/2$ . Our theorem crosses the Hill threshold $6/5$ but remains supercritical for Navier–Stokes.

On $\mathbb{T}^{3}$ , cutoff and mollification give a smooth compactly supported profile $\widetilde{H}_{R,e}$ . For

V^{p}(x,t)=A(t)\widetilde{H}_{R(t),e}(x-X(t)),

we introduce the Helmholtz correction

V^{c}=-\nabla\Delta^{-1}\operatorname{div}V^{p},\qquad V=V^{p}+V^{c}.

Then $V$ is divergence free. Choosing the intrinsic trajectory

X^{\prime}(t)=-\frac{A(t)}{R(t)}e,

cancels the leading transport and Euler terms. With

S_{R,e}:=\frac{1}{R}\widetilde{H}_{R,e},\qquad\int_{\mathbb{T}^{3}}S_{R,e}(x)\,\mathrm{d}x=-2\pi e.

The block then satisfies schematically

\partial_{t}V+\operatorname{div}(V\otimes V)-\nu\Delta V+\nabla P=\frac{\mathrm{d}}{\mathrm{d}t}\bigl(A(t)R(t)\bigr)S_{R(t),e}(x-X(t))+\operatorname{div}F,

(1.4)

where $F$ contains the remaining block errors.

The viscous contribution is written as the symmetric stress

-\nu\Delta V=\operatorname{div}\!\left[-\nu\bigl(\nabla V+(\nabla V)^{T}\bigr)\right].

It is estimated at an auxiliary exponent $1<p_{0}<6/5$ , where $2/p_{0}-5/3>0$ ; the limiting gradient is instead measured at $\bar{p}>6/5$ .

1.5 Strategy of the construction

A smooth Reynolds flow encodes the endpoint data, and an iteration removes its defect with localized Hill vortices moving along rational trajectories. At each stage, the stress is frozen on short time cells and decomposed into finitely many rank-one tensors. A moving Hill block treats each tensor, while a temporal corrector converts the resulting time-averaged cancellation into a pointwise identity.

The zeroth Reynolds flow and the iteration

The iteration is formulated for the Navier–Stokes–Reynolds system

\partial_{t}u_{q}+\operatorname{div}(u_{q}\otimes u_{q})-\nu\Delta u_{q}+\nabla p_{q}=\operatorname{div}\mathcal{E}_{q},\qquad\operatorname{div}u_{q}=0.

After mollifying $u_{q}$ , we add the Hill blocks and a temporal corrector. The principal blocks cancel the frozen part of $\mathcal{E}_{q}$ ; all remaining terms, including mollification, modulation, viscosity, and interactions with the coarse velocity, are placed in $\mathcal{E}_{q+1}$ . The estimates give $\mathcal{E}_{q}\to 0$ in $L_{t}^{\infty}L_{x}^{1}$ , summable $C_{t}L_{x}^{2}$ increments, and a common $C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}$ gradient bound. Hence

u\in C_{t}L_{x}^{2},\qquad\nabla u\in C_{t}L_{x}^{\bar{p}}.

The endpoint fields enter only at the initial stage. Given $u^{(0)},u^{(1)}\in L^{2}_{\sigma}(\mathbb{T}^{3})$ , we mollify them at a common scale $\ell$ and choose a smooth cutoff $\chi$ that equals one near $t=0$ and zero near $t=1$ . We then set

u_{0}(x,t):=\chi(t)(u^{(0)}*\rho_{\ell})(x)+(1-\chi(t))(u^{(1)}*\rho_{\ell})(x).

Its defect is placed in a symmetric Reynolds stress using $\mathcal{R}$ .

The blocks and corrector vanish at cell interfaces, including $t=0,1$ . Thus

\|u_{q+1}(\cdot,j)-u_{q}(\cdot,j)\|_{L^{2}}\leq C\lambda_{q+1}^{-\gamma}=C\lambda_{q}^{-1/5},\qquad j\in\{0,1\}.

These errors are summable. Suitable choices of $\ell$ and $\lambda_{0}$ then give the endpoint approximation.

Stress decomposition and moving Hill blocks

The geometric lemma decomposes the mollified stress, up to pressure, into nine positive rank-one pieces:

-\operatorname{div}\mathcal{E}_{\ell}=\operatorname{div}\left(\sum_{i=1}^{9}a_{i}(x,t)\xi_{i}\otimes\xi_{i}\right)+\nabla P^{d}.

The positivity of the coefficients permits them to be encoded in the radius and residence time of a Hill vortex. Each time cell is divided into nine disjoint subintervals, one for each direction, and the corresponding coefficient is frozen in time as $a_{i}^{k}(x)$ . This scheduling also removes the leading quadratic interactions between different directions.

Let

r_{q+1}:=\lambda_{q+1}^{-\mu},\qquad\mu>0,

be the base concentration parameter at stage $q+1$ ; the complete parameter hierarchy is stated in (1.6) below. For one coefficient $a=a_{i}^{k}$ and direction $\xi=\xi_{i}$ , set

R(x)=r_{q+1}a(x),\qquad A(t)=A_{0}\zeta(t),\qquad A_{0}^{2}=\frac{9}{2\pi}\int_{\mathbb{T}^{3}}a(x)\,\mathrm{d}x.

We orient the profile by $e=-\xi$ and define

V^{p}(x,t)=A(t)\widetilde{H}_{R(X(t)),-\xi}(x-X(t)).

The center follows

\dot{X}(t)=\frac{A(t)}{R(X(t))}\,\xi.

Thus $R=r_{q+1}a$ couples the stress coefficient to the Hill scale, whose physical core radius is $R^{2/3}$ , and the trajectory preserves the exact Hill cancellation. The normalization of $A_{0}$ compensates for the fact that the direction is active only during one ninth of the cell. Moreover, the speed is inversely proportional to $a$ : the vortex spends more time in regions where the coefficient is larger, which is the basis of the stress reconstruction below.

Reconstruction of the stress by time averages

The source in (1.4) is first replaced by a smooth auxiliary density of mass $2\pi$ . The difference has zero spatial mean and can therefore be written as a divergence with an inverse-frequency gain. Integration by parts along the trajectory converts the auxiliary source into a directional tensor flux. Since the speed is proportional to $(r_{q+1}a)^{-1}$ , the residence time recovers the coefficient $a$ , giving

\frac{1}{\tau}\int_{I_{i}}U(x,t)\,\mathrm{d}t=\operatorname{div}\bigl(a(x)\xi\otimes\xi+G(x)\bigr),\qquad\|G\|_{L^{1}}\lesssim\frac{\|a\|_{C^{1}}}{\lambda}.

(1.5)

A rational trajectory closes on the torus and can be repeated many times during one active interval. The freezing error is $O(\tau\|\partial_{t}a_{i}\|_{L^{\infty}})$ , while the $\lambda^{-1}$ factor controls incomplete periods and spatial variation; see Figure 3.

Figure 3: Schematic two-dimensional projection of the three-dimensional moving-core construction on

\mathbb{T}^{3}

(not to scale). In panel (a), the red background represents the frozen coefficient

a_{i}^{k}

. The dark inner balls show the vorticity core at scale

(R_{i}^{k})^{2/3}

, while the faint outer balls show the support of the principal localized profile at scale

(R_{i}^{k})^{\alpha}

, where

R_{i}^{k}=r_{q+1}a_{i}^{k}

. Panel (b) shows the resulting variable-width support trail; the disjointness condition is

2(R_{i}^{k})^{\alpha}\leq\lambda_{q+1}^{-1}

. Panel (c) shows the auxiliary density

\widetilde{U}_{i}^{k}(\,\cdot-x_{i}^{k}(t))

, whose support has the fixed scale

O(\lambda_{q+1}^{-1})

, rather than another velocity vortex. Panel (d) depicts its normalized full-orbit average, which is used in the approximate stress cancellation (1.5).

The temporal corrector and the new stress

The orbit identity cancels the stress only after averaging over a time cell. To restore the equation pointwise in time, sum the nine sources into $U_{\rm tot}$ and define

Q(x,t):=-\mathbb{P}\int_{k\tau}^{t}\left(U_{\rm tot}(x,s)-\frac{1}{\tau}\int_{J}U_{\rm tot}(x,z)\,\mathrm{d}z\right)\mathrm{d}s,\qquad t\in J,

where $\mathbb{P}$ is the Leray projection. Since the integrand has zero cell average, $Q$ vanishes at both endpoints and gains the short factor $\tau$ . Its time derivative cancels the zero-average source, while the cell average cancels the frozen stress through (1.5). The factor $\tau$ makes $Q$ smaller than the principal blocks in both the energy and gradient estimates.

The new stress contains the coarse-flow, corrector, freezing, source-replacement, localization, viscous, and intrinsic block errors. They are estimated separately because they use different features of the construction: concentration, inverse frequency, or the short time-cell factor. Their bounds make the stress decay and the increments summable, while endpoint vanishing preserves the prescribed traces.

1.6 Crossing the Hill threshold and analytical difficulties

Competing effects of concentration

The main difficulty is that concentration has opposite effects on the estimates. For an energy-sized Hill block,

\|\nabla H_{R,e}\|_{L^{p}}\lesssim R^{\frac{2}{p}-\frac{5}{3}}.

Thus small $R$ improves the intrinsic and viscous errors below $6/5$ , but increases the gradient norm above $6/5$ . We use $p_{0}<6/5$ for the block errors, $p_{s}>1$ close to one for source replacement, and $\bar{p}>6/5$ for the limiting gradient. Short time cells reduce the freezing and corrector errors, but must still contain enough complete rational orbits to average the stress.

The main parameters are

\lambda_{q+1}=\lambda_{q}^{\sigma},\qquad\delta_{q}=\lambda_{1}^{2\gamma}\lambda_{q}^{-\gamma},\qquad r_{q+1}=\lambda_{q+1}^{-\mu},\qquad\tau_{q+1}=\lambda_{q+1}^{-\eta}.

(1.6)

Here $\delta_{q}$ is the stress level, $r_{q}$ the concentration parameter, and $\tau_{q}$ the cell length. The rapid growth of $\lambda_{q}$ separates consecutive stages, whereas the small exponent $\gamma$ makes the stress decrease slowly enough to absorb the loss above the Hill threshold.

We take $p_{0}=28/27$ , for which $2/p_{0}-5/3=11/42>0$ . Together with the intrinsic block choices $\alpha=2/11$ and $\beta=502/231$ , this gives a positive power of the Hill scale in the intrinsic and viscous estimates. The source-replacement error is treated at $p_{s}>1$ close to one. The difference between the Hill source and the auxiliary source has zero mean, so a localized antidivergence produces an inverse-frequency gain; choosing $p_{s}$ near one keeps the associated concentration loss small.

The balance above $6/5$

For the final exponent, set

\kappa_{\ast}:=\frac{5}{3}-\frac{2}{\bar{p}}=\frac{5}{72003}>0.

A principal block has amplitude $\delta_{q+1}^{1/2}$ , while its smallest Hill parameter is $r_{q+1}\delta_{q+1}$ and its smallest physical core radius is $(r_{q+1}\delta_{q+1})^{2/3}$ . Therefore

\|\nabla V_{q+1}\|_{L_{t}^{\infty}L_{x}^{\bar{p}}}\lesssim\delta_{q+1}^{1/2}(r_{q+1}\delta_{q+1})^{-\kappa_{\ast}}.

Its summability requires

-\frac{\gamma}{2}+\kappa_{\ast}(\mu+\gamma)<0.

The factor $\delta_{q+1}^{1/2}$ must compensate for the concentration loss. This explains why increasing either $\bar{p}$ or $\mu$ makes the argument harder.

There is also a restriction from time averaging. Short cells improve the freezing error and the temporal primitive, but a cell must be long enough for the vortex to complete the required orbit periods. The orbit and temporal-corrector estimates impose the following sufficient conditions, respectively,

3-\mu+\frac{\gamma}{2}<-\eta,\qquad-\eta+4-\frac{3}{\bar{p}}+\frac{4n}{\sigma}+4\gamma<0.

These restrictions are compatible for

\sigma=200,\qquad n=20,\qquad\gamma=10^{-3},\qquad\mu=6+2\gamma=6.002,\qquad\eta=3,

and $\bar{p}=6/5+5\times 10^{-5}$ . This small gain above $6/5$ reflects the limited margin between the concentration loss and the orbit and corrector constraints. Temporal concentration cannot replace this balance: it improves time-integrated norms but does not reduce the required $L_{t}^{\infty}L_{x}^{\bar{p}}$ norm.

Localization, background interaction, and time continuity

Several further errors must be controlled. Localization and mollification create commutators, while the Helmholtz correction restores incompressibility after cutting off the exterior potential. The intrinsic Hill motion also produces an interaction with the coarse velocity. For a principal increment $v_{q+1}$ , concentration gives schematically

\|v_{q+1}\otimes u_{\ell}+u_{\ell}\otimes v_{q+1}\|_{L_{t}^{\infty}L_{x}^{1}}\lesssim\delta_{q+1}^{1/2}r_{q+1}^{1/3}\lambda_{q+1}^{\frac{2n}{\sigma}+\gamma}.

This must be smaller than the next stress level. The disjoint time scheduling removes the leading interactions between different principal directions, while orbit averaging, freezing, and the short temporal primitive control the remaining terms.

Finally, for some $\alpha_{0},c_{0}>0$ , the iteration provides estimates of the form

\|u_{q+1}-u_{q}\|_{C_{t}L_{x}^{2}}\lesssim\delta_{q+1}^{1/2},\qquad\|\nabla u_{q+1}\|_{C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}}\leq\|\nabla u_{q}\|_{C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}}+C\delta_{q+1}^{c_{0}}.

The first estimate gives strong convergence in $C_{t}L_{x}^{2}$ ; the second provides a common time modulus for the gradients. Together they yield $u\in C_{t}L_{x}^{2}$ and $\nabla u\in C_{t}L_{x}^{\bar{p}}$ . A uniform fixed-time gradient bound alone would not give this time continuity.

1.7 Organization of the paper

The remainder of the paper is organized as follows. Section 2 recalls the normalized Hill vortex and establishes estimates for its rescaled, localized, and mollified forms. Section 3 introduces the moving Hill block and derives the associated Euler and Navier–Stokes defect identities. Section 4 states the iteration step and constructs the next Reynolds flow, including the decomposition of the Reynolds stress, the time partition, the space-dependent scales, the vortex trajectories, the auxiliary source, and the temporal corrector. Section 5 proves the estimates required for the iteration and verifies the parameter restrictions. Section 6 constructs the initial Reynolds flow, passes to the limit, and proves Theorems 1.1 and 1.2. Section 7 discusses the limitations of the method and possible modifications. Section 8 presents further directions and open problems.

2 Hill’s spherical vortex

Hill’s spherical vortex is an axisymmetric, no-swirl solution of the three-dimensional Euler equations whose vorticity is supported in a ball and which translates without changing shape. It was introduced by Hill [60]; see also Lamb [73, §165] for the classical formulas and Choi [33] for a modern treatment of its orbital stability in the axisymmetric no-swirl class. Only the explicit traveling-wave structure is used below.

We record the profile carefully because three features enter the convex-integration scheme with their precise normalization: the translation speed, the directional integral of the localized velocity, and the potential character of the exterior field. We first write the classical vortex in a frame in which the spherical core is stationary, then remove the constant velocity at infinity to obtain a decaying traveling profile. The following subsection rescales and rotates this unit profile and truncates its exterior potential without changing the vorticity core. Here $\xi_{i}$ denotes a rank-one stress direction. Also, $Q_{e}\in SO(3)$ denotes a rotation, whereas $Q_{q+1}$ is the temporal corrector introduced in Section 4.

Let $(r,\theta,\varphi)$ be spherical coordinates with the axis of propagation along the $z$ -axis. In this preliminary description, $r=|x|$ denotes the spherical radius; it should not be confused with the concentration parameters introduced later. Thus

z=r\cos\theta,\qquad s=r\sin\theta,

where $s$ is the cylindrical radius. The flow is axisymmetric and has no swirl; hence

u_{\varphi}=0.

An axisymmetric no-swirl incompressible velocity is represented by its Stokes stream function $\psi(r,\theta)$ , with

u_{r}=\frac{1}{r^{2}\sin\theta}\,\partial_{\theta}\psi,\qquad u_{\theta}=-\frac{1}{r\sin\theta}\,\partial_{r}\psi.

For a Hill vortex of radius $a$ and translation speed $U$ , we use the sign convention

\psi(r,\theta)=\begin{cases}-\dfrac{3U}{4}\left(1-\dfrac{r^{2}}{a^{2}}\right)r^{2}\sin^{2}\theta,&r\leq a,\\[5.16663pt] \dfrac{U}{2}\left(1-\dfrac{a^{3}}{r^{3}}\right)r^{2}\sin^{2}\theta,&r\geq a.\end{cases}

These formulas agree, up to the choice of propagation direction, with the classical formulas in [60]; see also [73, §165] for the normalization used here.

The corresponding velocity components are

u_{r}=\begin{cases}-\dfrac{3U}{2}\left(1-\dfrac{r^{2}}{a^{2}}\right)\cos\theta,&r\leq a,\\[5.16663pt] U\left(1-\dfrac{a^{3}}{r^{3}}\right)\cos\theta,&r\geq a,\end{cases}

u_{\theta}=\begin{cases}\dfrac{3U}{2}\left(1-\dfrac{2r^{2}}{a^{2}}\right)\sin\theta,&r\leq a,\\[5.16663pt] -\,U\left(1+\dfrac{a^{3}}{2r^{3}}\right)\sin\theta,&r\geq a,\end{cases}

u_{\varphi}=0.

The vorticity is purely azimuthal:

\omega=\omega_{\varphi}\,e_{\varphi},

with

\omega_{\varphi}=\begin{cases}-\dfrac{15U}{2a^{2}}\,r\sin\theta,&r\leq a,\\[5.16663pt] 0,&r\geq a.\end{cases}

In particular, in the normalization used here, the relative vorticity is

\frac{\omega_{\varphi}}{r\sin\theta}=-\frac{15U}{2a^{2}}\mathbf{1}_{B_{a}}.

Thus the vorticity is compactly supported even though the induced velocity has a noncompact potential tail.

These formulas describe a stationary field in the co-moving frame. In that frame the velocity is an exact steady solution of the incompressible Euler equations,

(u\cdot\nabla)u+\nabla p=0,\qquad\nabla\cdot u=0.

For $U=a=1$ , the co-moving velocity $u$ tends to $e_{3}$ at infinity. Subtracting this constant far-field velocity gives the decaying laboratory-frame profile

H:=u-e_{3}.

Since $u=H+e_{3}$ , the steady Euler equation becomes, in the sense of distributions,

\,\partial_{3}H+(H\cdot\nabla)H+\nabla P_{H}=0,\qquad\operatorname{div}H=0.

Thus $H(x+te_{3})$ travels in the direction $-e_{3}$ .

In spherical coordinates, let $\hat{r}$ , $\hat{\theta}$ , and $\hat{\varphi}$ denote the radial, polar, and azimuthal unit vectors, respectively. Since $H$ has no azimuthal component, it is determined by

H\cdot\hat{r}=\begin{cases}-\dfrac{3}{2}\left(\dfrac{5}{3}-r^{2}\right)\cos\theta,&r\leq 1,\\[5.16663pt] -\dfrac{1}{r^{3}}\cos\theta,&r\geq 1,\end{cases}

H\cdot\hat{\theta}=\begin{cases}\dfrac{3}{2}\left(\dfrac{5}{3}-2r^{2}\right)\sin\theta,&r\leq 1,\\[5.16663pt] -\dfrac{1}{2r^{3}}\sin\theta,&r\geq 1.\end{cases}

For $r\geq 1$ , the profile $H$ coincides with the three-dimensional potential doublet $H=\nabla\Phi$ , where $\Phi(x)=z/(2r^{3})$ .

	$\displaystyle H$	$\displaystyle=-\frac{1}{r^{3}}\cos\theta\begin{pmatrix}\sin\theta\cos\varphi\\ \sin\theta\sin\varphi\\ \cos\theta\end{pmatrix}-\frac{1}{2r^{3}}\sin\theta\begin{pmatrix}\cos\theta\cos\varphi\\ \cos\theta\sin\varphi\\ -\sin\theta\end{pmatrix}$
		$\displaystyle=-\frac{1}{r^{5}}\begin{pmatrix}\frac{3}{2}xz\\ \frac{3}{2}yz\\ z^{2}-\frac{1}{2}(x^{2}+y^{2})\end{pmatrix}$
		$\displaystyle=\nabla\!\left(\frac{z}{2r^{3}}\right)=\nabla\Phi,\qquad r\geq 1.$

Remark 2.1.

In Cartesian coordinates, $H$ is given by

H=\begin{cases}-\begin{pmatrix}\frac{3}{2}xz\\ \frac{3}{2}yz\\ -3(x^{2}+y^{2})-\frac{3}{2}z^{2}+\frac{5}{2}\end{pmatrix}&r\leq 1,\\ -\frac{1}{r^{5}}\begin{pmatrix}\frac{3}{2}xz\\ \frac{3}{2}yz\\ z^{2}-\frac{1}{2}(x^{2}+y^{2})\end{pmatrix}&r\geq 1.\end{cases}

Thus $H\in C^{0,1}(\mathbb{R}^{3})$ and is smooth on $\mathbb{R}^{3}\setminus\partial B_{1}(0)$ .

We shall use four consequences of these formulas. The field $H$ is a finite-energy divergence-free traveling Euler profile; its vorticity is confined to $B_{1}$ ; its exterior velocity is the gradient $\nabla\Phi$ ; and its only loss of smoothness occurs across $\partial B_{1}$ . The potential representation permits us to remove the long-range tail by a cutoff. Since the profile is only Lipschitz across the spherical interface, we subsequently mollify it to obtain the required higher-derivative estimates.

2.1 Scaled Hill vortex profile

We first introduce the spatial and amplitude scalings used in the construction. The spatial scale of the vorticity core is $R^{2/3}$ , whereas the amplitude is $R^{-1}$ . This preserves the $L^{2}$ -size of the profile and changes its traveling speed from one to $R^{-1}$ . We then cut off the irrotational exterior field at the larger scale $R^{\alpha}$ . Since $\alpha<2/3$ , this localization does not alter the vorticity core.

For $R>0$ , define the rescaled Hill profile by

H_{R}(x):=\frac{1}{R}\,H\!\left(\frac{x}{R^{\frac{2}{3}}}\right),\qquad P_{R}(x):=\frac{1}{R^{2}}\,P_{H}\!\left(\frac{x}{R^{\frac{2}{3}}}\right).

Then

\frac{1}{R}\,\partial_{3}H_{R}+\operatorname{div}(H_{R}\otimes H_{R})+\nabla P_{R}=0,\qquad\operatorname{div}H_{R}=0.

(2.1)

More generally, for $e\in\mathbb{S}^{2}$ , choose $Q_{e}\in SO(3)$ with $Q_{e}e_{3}=e$ , and define

H_{R,e}(x):=Q_{e}H_{R}(Q_{e}^{T}x),\qquad P_{R,e}(x):=P_{R}(Q_{e}^{T}x).

The definition is independent of the particular choice of $Q_{e}$ , because the Hill profile is axisymmetric about $e_{3}$ . The field $H_{R,e}$ is oriented along $e$ , travels intrinsically in the direction $-e$ , and satisfies

\frac{1}{R}(e\cdot\nabla)H_{R,e}+\operatorname{div}(H_{R,e}\otimes H_{R,e})+\nabla P_{R,e}=0,\qquad\operatorname{div}H_{R,e}=0.

Equivalently,

(x,t)\longmapsto H_{R,e}\!\left(x+\frac{t}{R}e\right)

is a traveling Euler solution whose center moves with velocity $-R^{-1}e$ .

Fix $0<R\leq R_{0}<1$ and $0<\alpha<2/3$ . Let $\chi\in C^{\infty}(\mathbb{R}^{3})$ be radial, with $\chi=0$ on $B_{1}$ , $\chi=1$ on $\mathbb{R}^{3}\setminus B_{2}$ , and $0\leq\chi\leq 1$ . Set

\chi_{\alpha}(x):=\chi\!\left(\frac{x}{R^{\alpha}}\right).

Since $R^{2/3}<R^{\alpha}$ , the transition annulus

A_{R}:=B_{2R^{\alpha}}(0)\setminus\overline{B}_{R^{\alpha}}(0)

lies strictly outside the spherical vorticity core. For $|x|\geq R^{2/3}$ , the rescaled profile is a gradient. Indeed, the $(-3)$ -homogeneity of $\nabla\Phi$ gives

H_{R}=\frac{1}{R}\,H\!\left(\frac{x}{R^{\frac{2}{3}}}\right)=\frac{1}{R}\,\nabla\Phi\!\left(\frac{x}{R^{\frac{2}{3}}}\right)=R\nabla\Phi(x).

The scalar potential and the localized Hill profile are therefore defined by

\Pi_{R}:=R\chi_{\alpha}\Phi,\qquad\bar{H}_{R}:=H_{R}-\nabla\Pi_{R}=H_{R}-R\nabla(\chi_{\alpha}\Phi).

(2.2)

Although $\Phi$ is singular at the origin, $\Pi_{R}$ is smooth because $\chi_{\alpha}$ vanishes in a neighborhood of the origin. Thus $\bar{H}_{R}=H_{R}$ on $B_{R^{\alpha}}$ , while $\bar{H}_{R}=0$ on $\mathbb{R}^{3}\setminus B_{2R^{\alpha}}$ . In particular, the gradient correction preserves the vorticity exactly:

\nabla\times\bar{H}_{R}=\nabla\times H_{R}.

It introduces only a divergence error in the annulus $A_{R}$ . Since $\Delta\Phi=0$ on $\mathbb{R}^{3}\setminus\{0\}$ ,

\operatorname{div}\bar{H}_{R}=-R\Delta(\chi_{\alpha}\Phi)=-R(\Delta\chi_{\alpha})\Phi-2R\nabla\chi_{\alpha}\cdot\nabla\Phi,\qquad\operatorname{supp}\operatorname{div}\bar{H}_{R}\subseteq\overline{A}_{R}.

Remark 2.2.

As shown in Lemma 2.3, the scaling and radial localization give

\frac{1}{R}\int_{\mathbb{R}^{3}}\bar{H}_{R}(x)\,\mathrm{d}x=-2\pi e_{3},

independently of $R$ . The same scaling produces the factor $R^{-1}$ in the traveling equation and in the trajectory equation of Proposition 3.1.

2.2 Estimates for the building block

We collect the estimates for the localized profile. The first lemma records its support, Lebesgue norms, derivative bounds, and directional integral. We then derive the approximate traveling equation and the identity obtained by differentiating in $R$ , and finally mollify the spherical interface.

Lemma 2.3.

The fields $\bar{H}_{R}$ and $\Pi_{R}$ , defined in (2.2), satisfy the following properties, with constants independent of $R\in(0,R_{0}]$ . The constants may depend on $\alpha$ , $R_{0}$ , and the cutoff $\chi$ , and also on the displayed Lebesgue exponent.

The localization identities are

\bar{H}_{R}=H_{R},\quad\Pi_{R}=0\quad\text{on }B_{R^{\alpha}}(0),\qquad\bar{H}_{R}=0,\quad\nabla\Pi_{R}=H_{R}\quad\text{on }\mathbb{R}^{3}\setminus B_{2R^{\alpha}}(0).

Consequently, $\operatorname{supp}\bar{H}_{R}\subseteq\overline{B}_{2R^{\alpha}}(0)$ and $\operatorname{supp}\operatorname{div}\bar{H}_{R}\subseteq\overline{A}_{R}$ .

2.

$\|\bar{H}_{R}\|_{L^{1}}\leq CR(1+|\log R|)$ and $\|\bar{H}_{R}\|_{L^{p}}\leq CR^{\frac{2}{p}-1}$ for every $1<p\leq\infty$ .

For every $1\leq p\leq\infty$ , $\|\nabla\bar{H}_{R}\|_{L^{p}}\leq CR^{\frac{2}{p}-\frac{5}{3}}$ ,

\|\nabla^{2}\bar{H}_{R}\|_{L^{p}(\mathbb{R}^{3}\setminus\partial B_{R^{2/3}})}\leq CR^{\frac{2}{p}-\frac{7}{3}},\qquad\|\operatorname{div}\bar{H}_{R}\|_{L^{p}}\leq CR^{1-4\alpha+\frac{3\alpha}{p}}.

4.

$\frac{1}{R}\int_{\mathbb{R}^{3}}\bar{H}_{R}(x)\,\mathrm{d}x=\begin{pmatrix}0\\ 0\\ -2\pi\end{pmatrix}.$ (2.3)

Proof.

The identities in Part 1 follow from the definition of the cutoff and the exterior formula $H_{R}=R\nabla\Phi$ .

For the norm estimates, the explicit formulas for $H$ and $\Phi$ give, for $n=0,1,2$ ,

|\nabla^{n}H_{R}(x)|\leq\begin{cases}CR^{-1-\frac{2n}{3}},&|x|<R^{2/3},\\ CR|x|^{-n-3},&|x|>R^{2/3},\end{cases}

(2.4)

where the estimate for $n=2$ is understood away from the interface. Moreover, on the support of a derivative of $\chi_{\alpha}$ one has $|x|\simeq R^{\alpha}$ . Hence, for $n=0,1,2$ ,

|\nabla^{n}\Pi_{R}(x)|\leq CR|x|^{-n-2},\qquad x\neq 0.

(2.5)

Combining (2.4) and (2.5), we obtain

|\nabla^{n}\bar{H}_{R}(x)|\leq\begin{cases}CR^{-1-\frac{2n}{3}},&|x|<R^{2/3},\\ CR|x|^{-n-3},&R^{2/3}<|x|<2R^{\alpha},\end{cases}

(2.6)

again with $n=2$ understood away from $\partial B_{R^{2/3}}$ . Integrating (2.6) in polar coordinates proves the $L^{p}$ -bounds in Parts 2 and 3. At $p=1$ , the exterior contribution to $\|\bar{H}_{R}\|_{L^{1}}$ is $CR\int_{R^{2/3}}^{2R^{\alpha}}r^{-1}\,\mathrm{d}r$ , which accounts for the factor $1+|\log R|$ .

It remains to estimate the divergence. On $A_{R}$ , $|\nabla^{k}\chi_{\alpha}|\leq CR^{-k\alpha}$ , while $|\Phi|\leq CR^{-2\alpha}$ and $|\nabla\Phi|\leq CR^{-3\alpha}$ . Thus

|\operatorname{div}\bar{H}_{R}|\leq CR^{1-4\alpha}\mathbf{1}_{A_{R}},

which gives $\|\operatorname{div}\bar{H}_{R}\|_{L^{p}}\leq CR^{1-4\alpha+3\alpha/p}$ . The warning concerning the second derivative is essential: $H$ is Lipschitz across the spherical interface, but it need not have a classical second derivative there. The mollification below removes this singularity.

Finally, compact support of $\bar{H}_{R}$ and integration by parts give, componentwise,

	$\displaystyle\frac{1}{R}\int_{\mathbb{R}^{3}}\bar{H}_{R}(x)\,\mathrm{d}x$	$\displaystyle=-\frac{1}{R}\int_{\mathbb{R}^{3}}x\,\operatorname{div}\bar{H}_{R}(x)\,\mathrm{d}x$
		$\displaystyle=\int_{\mathbb{R}^{3}}x\bigl((\Delta\chi_{\alpha})\Phi+2\nabla\chi_{\alpha}\cdot\nabla\Phi\bigr)\,\mathrm{d}x$
		$\displaystyle=-\int_{\mathbb{R}^{3}}\Phi\nabla\chi_{\alpha}\,\mathrm{d}x+\int_{\mathbb{R}^{3}}x(\nabla\chi_{\alpha}\cdot\nabla\Phi)\,\mathrm{d}x.$

Write $\chi_{\alpha}^{\prime}(r)=\frac{\mathrm{d}}{\mathrm{d}r}\chi(r/R^{\alpha})$ . Since $\Phi=\cos\theta/(2r^{2})$ and $\partial_{r}\Phi=-\cos\theta/r^{3}$ , the last line equals

-\frac{3}{2}\left(\int_{0}^{\infty}\chi_{\alpha}^{\prime}(r)\,\mathrm{d}r\right)\int_{0}^{2\pi}\!\int_{0}^{\pi}\cos\theta\,\widehat{r}\,\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi.

The radial integral is one. The first two angular components vanish, and the third is

2\pi\int_{0}^{\pi}\cos^{2}\theta\sin\theta\,\mathrm{d}\theta=\frac{4\pi}{3}.

This proves (2.3). ∎

The next proposition shows that the truncated block $\bar{H}_{R}$ satisfies the traveling Euler-profile equation up to error terms. We also compute $\partial_{R}\bar{H}_{R}$ , because the scale varies along the moving trajectory. We use the following Bogovskiĭ notation; see [9, 51] and, for the scaling-invariant $L^{p}$ bound used below, [1]. If $f=(f_{1},f_{2},f_{3})$ is a vector field supported in a ball $B_{r}$ and $\int_{B_{r}}f_{i}\,\mathrm{d}x=0$ for each $i$ , we define the componentwise Bogovskiĭ tensor by

(\mathcal{B}_{r}f)_{ij}:=(\mathcal{B}_{r}f_{i})_{j}.

It is extended by zero outside $B_{r}$ and satisfies

\operatorname{div}\mathcal{B}_{r}f=f,\qquad\operatorname{supp}\mathcal{B}_{r}f\subseteq\overline{B}_{r},\qquad\|\mathcal{B}_{r}f\|_{L^{p}}\leq Cr\|f\|_{L^{p}},\quad 1<p<\infty.

This componentwise tensor is not required to be symmetric; symmetry is restored later by the symmetric inverse-divergence operator on the torus.

Proposition 2.4.

Let $\alpha\in(0,2/3)$ . Then the truncated block $\bar{H}_{R}$ satisfies the following identities in the sense of distributions on $\mathbb{R}^{3}$ .

\frac{1}{R}\,\partial_{3}\bar{H}_{R}+\operatorname{div}(\bar{H}_{R}\otimes\bar{H}_{R})+\nabla P_{1}=\operatorname{div}F_{1},

(2.7)

\partial_{R}\bar{H}_{R}=\frac{1}{R}\bar{H}_{R}+\nabla P_{2}+\operatorname{div}F_{2},

(2.8)

where the error and pressure terms have the following properties:

After fixing the additive constant in $P_{1}$ ,

\operatorname{supp}P_{1},\ \operatorname{supp}P_{2},\ \operatorname{supp}F_{1},\ \operatorname{supp}F_{2}\subseteq\overline{B}_{2R^{\alpha}}(0).

2.

For every $p\in(1,\infty)$ , $\|F_{1}\|_{L^{p}}\leq CR^{2-6\alpha+3\alpha/p}$ and $\|F_{2}\|_{L^{p}}\leq CR^{2/p-4/3}$ .

\|\nabla P_{2}\|_{L^{\infty}}\leq CR^{-3\alpha},\qquad\|\nabla^{2}P_{2}\|_{L^{\infty}}\leq CR^{-4\alpha},\qquad\|\operatorname{div}F_{2}\|_{L^{\infty}}\leq CR^{-2},

and, away from the spherical interface,

\|\nabla\operatorname{div}F_{2}\|_{L^{\infty}(\mathbb{R}^{3}\setminus\partial B_{R^{2/3}})}\leq CR^{-8/3}.

Proof.

The calculation follows the localization mechanism in [14, Proposition 2.1]. Since $\bar{H}_{R}=H_{R}-\nabla\Pi_{R}$ and $H_{R}$ satisfies (2.1), we have

\frac{1}{R}\,\partial_{3}\bar{H}_{R}+\frac{1}{R}\,\partial_{3}\nabla\Pi_{R}+\operatorname{div}(\bar{H}_{R}\otimes\bar{H}_{R})+\operatorname{div}(\nabla\Pi_{R}\otimes\bar{H}_{R})\\ +\operatorname{div}(\bar{H}_{R}\otimes\nabla\Pi_{R})+\operatorname{div}(\nabla\Pi_{R}\otimes\nabla\Pi_{R})+\nabla P_{R}=0.

We use the identity

\operatorname{div}(\nabla\Pi_{R}\otimes\nabla\Pi_{R})=(\Delta\Pi_{R})\nabla\Pi_{R}+\frac{1}{2}\nabla|\nabla\Pi_{R}|^{2}.

The vector field $g_{R}:=(\Delta\Pi_{R})\nabla\Pi_{R}$ is supported in $\overline{A}_{R}$ . To apply $\mathcal{B}_{2R^{\alpha}}$ , we verify that it has zero mean. The identity

g_{R}=\operatorname{div}\!\left(\nabla\Pi_{R}\otimes\nabla\Pi_{R}-\frac{1}{2}|\nabla\Pi_{R}|^{2}I\right)

and the exterior decay $\nabla\Pi_{R}=R\nabla\Phi=O(R|x|^{-3})$ show, by integration over $B_{L}$ and passage to the limit $L\to\infty$ , that

\int_{\mathbb{R}^{3}}g_{R}\,\mathrm{d}x=0.

Indeed, the boundary flux is $O(R^{2}L^{-4})$ . Thus every component of $g_{R}$ satisfies the compatibility condition for the componentwise Bogovskiĭ operator.

Equation (2.7) for the truncated block follows upon setting

P_{1}=P_{R}+\frac{1}{R}\partial_{3}\Pi_{R}+\frac{1}{2}|\nabla\Pi_{R}|^{2},

F_{1}=-\nabla\Pi_{R}\otimes\bar{H}_{R}-\bar{H}_{R}\otimes\nabla\Pi_{R}-\mathcal{B}_{2R^{\alpha}}g_{R}.

(2.9)

On the exterior of $B_{2R^{\alpha}}$ , we have $\nabla\Pi_{R}=H_{R}$ and $\Delta\Pi_{R}=0$ . The potential-flow form of (2.1) therefore implies $\nabla P_{1}=0$ there. We fix the additive constant so that $P_{1}=0$ in the exterior. The stated support properties of $P_{1}$ and $F_{1}$ now follow from the support-preserving property of $\mathcal{B}_{2R^{\alpha}}$ .

The support of $\nabla\Pi_{R}\otimes\bar{H}_{R}$ lies in $\overline{A}_{R}$ by Lemma 2.3. On this annulus,

|\bar{H}_{R}|+|\nabla\Pi_{R}|\leq CR^{1-3\alpha},\qquad|\nabla^{2}\Pi_{R}|\leq CR^{1-4\alpha}.

Since $|A_{R}|\leq CR^{3\alpha}$ , it follows that

	$\displaystyle\\|\nabla\Pi_{R}\otimes\bar{H}_{R}+\bar{H}_{R}\otimes\nabla\Pi_{R}\\|_{L^{p}}$	$\displaystyle\leq CR^{2-6\alpha+\frac{3\alpha}{p}},$
	$\displaystyle\\|g_{R}\\|_{L^{p}}$	$\displaystyle\leq CR^{2-7\alpha+\frac{3\alpha}{p}}.$

The scale- $R^{\alpha}$ Bogovskiĭ estimate then yields

\|\mathcal{B}_{2R^{\alpha}}g_{R}\|_{L^{p}}\leq CR^{\alpha}\|g_{R}\|_{L^{p}}\leq CR^{2-6\alpha+\frac{3\alpha}{p}}.

Combining the preceding estimates with (2.9) proves the bound for $F_{1}$ .

Next, we compute $\partial_{R}H_{R}$ .

\partial_{R}H_{R}=-\frac{1}{R}H_{R}-\frac{2}{3}\frac{x}{R}\cdot\nabla H_{R}=\frac{1}{R}H_{R}-\frac{2}{3}\operatorname{div}\!\left(H_{R}\otimes\frac{x}{R}\right).

Differentiating the cutoff term as well gives

\partial_{R}\bar{H}_{R}=\frac{1}{R}\bar{H}_{R}-\nabla(R\Phi\,\partial_{R}\chi_{\alpha})-\frac{2}{3}D_{R},\qquad D_{R}:=\operatorname{div}\!\left(H_{R}\otimes\frac{x}{R}\right).

(2.10)

In the exterior region, $H_{R}=R\nabla\Phi$ is homogeneous of degree $-3$ , and therefore $D_{R}=0$ for $|x|>R^{2/3}$ . Thus $\operatorname{supp}D_{R}\subseteq\overline{B}_{R^{2/3}}$ . Integrating (2.10) and using (2.3), we also obtain

\int_{\mathbb{R}^{3}}D_{R}\,\mathrm{d}x=-\frac{3}{2}R\,\partial_{R}\!\left(\frac{1}{R}\int_{\mathbb{R}^{3}}\bar{H}_{R}\,\mathrm{d}x\right)=0.

We may therefore set

P_{2}:=-R\Phi\,\partial_{R}\chi_{\alpha},\qquad F_{2}:=-\frac{2}{3}\mathcal{B}_{R^{2/3}}D_{R}.

This proves (2.8). From (2.4),

\|D_{R}\|_{L^{p}}\leq CR^{-2+\frac{2}{p}},\qquad\|D_{R}\|_{L^{\infty}}\leq CR^{-2},\qquad\|\nabla D_{R}\|_{L^{\infty}(\mathbb{R}^{3}\setminus\partial B_{R^{2/3}})}\leq CR^{-8/3}.

The scale- $R^{2/3}$ Bogovskiĭ estimate gives

\|F_{2}\|_{L^{p}}\leq CR^{2/3}\|D_{R}\|_{L^{p}}\leq CR^{\frac{2}{p}-\frac{4}{3}}.

Finally,

\partial_{R}\chi_{\alpha}(x)=-\frac{\alpha}{R}\left(\frac{x}{R^{\alpha}}\cdot\nabla\chi\!\left(\frac{x}{R^{\alpha}}\right)\right)

is supported in $\overline{A}_{R}$ . The bounds for $P_{2}$ , $\operatorname{div}F_{2}=-\frac{2}{3}D_{R}$ , and its piecewise gradient now follow directly. ∎

Corollary 2.5 (Mollified Hill block).

Fix $p\in(1,\infty)$ and a radial function $\rho\in C_{c}^{\infty}(B_{1})$ such that $\rho\geq 0$ and $\int_{\mathbb{R}^{3}}\rho\,\mathrm{d}x=1$ . Set

\beta:=\frac{14}{3}+\frac{3\alpha-2}{p}-6\alpha,\qquad\rho_{R^{\beta}}(x):=R^{-3\beta}\rho(x/R^{\beta}),\qquad\widetilde{H}_{R}:=\rho_{R^{\beta}}\ast\bar{H}_{R}.

Then $\beta>2/3$ , and there exist $\widetilde{P}_{1},\widetilde{P}_{2},\widetilde{F}_{1},\widetilde{F}_{2}$ such that

	$\displaystyle\frac{1}{R}\partial_{3}\widetilde{H}_{R}+\operatorname{div}(\widetilde{H}_{R}\otimes\widetilde{H}_{R})+\nabla\widetilde{P}_{1}$	$\displaystyle=\operatorname{div}\widetilde{F}_{1},$		(2.11)
	$\displaystyle\partial_{R}\widetilde{H}_{R}$	$\displaystyle=\frac{1}{R}\widetilde{H}_{R}+\nabla\widetilde{P}_{2}+\operatorname{div}\widetilde{F}_{2}.$		(2.12)

Moreover,

\|\widetilde{F}_{1}\|_{L^{p}}\leq CR^{2-6\alpha+\frac{3\alpha}{p}},\qquad\|\widetilde{F}_{2}\|_{L^{p}}\leq CR^{\frac{2}{p}-\frac{4}{3}},

and

\|\nabla\widetilde{P}_{2}\|_{L^{\infty}}\leq CR^{-3\alpha},\qquad\|\nabla^{2}\widetilde{P}_{2}\|_{L^{\infty}}\leq CR^{-4\alpha},\qquad\|\operatorname{div}\widetilde{F}_{2}\|_{L^{\infty}}\leq CR^{-2},

\|\nabla\operatorname{div}\widetilde{F}_{2}\|_{L^{\infty}}\leq CR^{-2-\beta}.

(2.13)

For every $1<q\leq\infty$ , the mollified profile also satisfies

\|\widetilde{H}_{R}\|_{L^{q}}\leq CR^{\frac{2}{q}-1},\qquad\|\nabla\widetilde{H}_{R}\|_{L^{q}}\leq CR^{\frac{2}{q}-\frac{5}{3}},\qquad\|\nabla^{2}\widetilde{H}_{R}\|_{L^{q}}\leq CR^{\frac{2}{q}-\frac{5}{3}-\beta},

and

\|\operatorname{div}\widetilde{H}_{R}\|_{L^{q}}\leq CR^{1-4\alpha+\frac{3\alpha}{q}}.

At $q=1$ , the corresponding endpoint estimates are

\|\widetilde{H}_{R}\|_{L^{1}}\leq CR(1+|\log R|),\qquad\|\nabla\widetilde{H}_{R}\|_{L^{1}}\leq CR^{1/3},

\|\nabla^{2}\widetilde{H}_{R}\|_{L^{1}}\leq CR^{1/3-\beta},\qquad\|\operatorname{div}\widetilde{H}_{R}\|_{L^{1}}\leq CR^{1-\alpha}.

The fields $\widetilde{H}_{R}$ , $\widetilde{P}_{1}$ , $\widetilde{P}_{2}$ , $\widetilde{F}_{1}$ , and $\widetilde{F}_{2}$ are supported in $\overline{B}_{3R^{\alpha}}(0)$ , and

\frac{1}{R}\int_{\mathbb{R}^{3}}\widetilde{H}_{R}\,\mathrm{d}x=-2\pi e_{3}.

Proof.

Mollifying (2.7) in space gives (2.11) with

\widetilde{P}_{1}:=P_{1}\ast\rho_{R^{\beta}}

and

\widetilde{F}_{1}:=F_{1}\ast\rho_{R^{\beta}}+\widetilde{H}_{R}\otimes\widetilde{H}_{R}-(\bar{H}_{R}\otimes\bar{H}_{R})\ast\rho_{R^{\beta}}.

Young’s convolution inequality gives $\|\rho_{R^{\beta}}\ast F_{1}\|_{L^{p}}\leq CR^{2-6\alpha+3\alpha/p}$ . Set

\mathcal{C}_{R}:=(\bar{H}_{R}\otimes\bar{H}_{R})\ast\rho_{R^{\beta}}-(\bar{H}_{R}\ast\rho_{R^{\beta}})\otimes(\bar{H}_{R}\ast\rho_{R^{\beta}}).

Then

	$\displaystyle\\|\mathcal{C}_{R}\\|_{L^{p}}$	$\displaystyle\leq CR^{\beta}\\|\bar{H}_{R}\\|_{L^{2p}}\\|\nabla\bar{H}_{R}\\|_{L^{2p}}$
		$\displaystyle\leq CR^{\beta+\frac{2}{p}-\frac{8}{3}}=CR^{2-6\alpha+\frac{3\alpha}{p}}.$

Here we used precisely the stated choice of $\beta$ . Since $3\alpha-2<0$ and $p>1$ ,

\beta>\frac{14}{3}+(3\alpha-2)-6\alpha=\frac{8}{3}-3\alpha>\frac{2}{3}.

This proves the estimate for $\widetilde{F}_{1}$ .

We next compute the derivative with respect to $R$ . Since the mollification scale depends on $R$ ,

\partial_{R}\widetilde{H}_{R}=(\partial_{R}\bar{H}_{R})\ast\rho_{R^{\beta}}+\bar{H}_{R}\ast\partial_{R}\rho_{R^{\beta}}.

Mollifying (2.8) gives

(\partial_{R}\bar{H}_{R})\ast\rho_{R^{\beta}}=\frac{1}{R}\widetilde{H}_{R}+\nabla(P_{2}\ast\rho_{R^{\beta}})+\operatorname{div}(F_{2}\ast\rho_{R^{\beta}}).

For vector fields $F,G$ , define the second-order tensor $F\circledast G$ by $(F\circledast G)_{ij}:=F_{i}\ast G_{j}$ , and set

K_{R}(x):=\frac{x}{R}\rho_{R^{\beta}}(x).

A direct differentiation of the rescaled kernel gives

\partial_{R}\rho_{R^{\beta}}=-\frac{\beta}{R}\left(3\rho_{R^{\beta}}+x\cdot\nabla\rho_{R^{\beta}}\right)=-\beta\,\operatorname{div}K_{R}.

Consequently,

\bar{H}_{R}\ast\partial_{R}\rho_{R^{\beta}}=-\beta\,\operatorname{div}(\bar{H}_{R}\circledast K_{R}).

Thus (2.12) holds with

\widetilde{P}_{2}:=P_{2}\ast\rho_{R^{\beta}},\qquad\widetilde{F}_{2}:=F_{2}\ast\rho_{R^{\beta}}-\beta\,\bar{H}_{R}\circledast K_{R}.

The kernel satisfies $\|K_{R}\|_{L^{1}}\leq CR^{\beta-1}$ and $\|\nabla K_{R}\|_{L^{1}}\leq CR^{-1}$ . Hence

\displaystyle\|\bar{H}_{R}\circledast K_{R}\|_{L^{p}}\leq C\|K_{R}\|_{L^{1}}\|\bar{H}_{R}\|_{L^{p}}\leq CR^{\beta+\frac{2}{p}-2}\leq CR^{\frac{2}{p}-\frac{4}{3}}.

We used $\beta>2/3$ in the last inequality. Similarly,

	$\displaystyle\\|\operatorname{div}(\bar{H}_{R}\circledast K_{R})\\|_{L^{\infty}}$	$\displaystyle\leq C\\|K_{R}\\|_{L^{1}}\\|\nabla\bar{H}_{R}\\|_{L^{\infty}}\leq CR^{\beta-\frac{8}{3}}\leq CR^{-2},$
	$\displaystyle\\|\nabla\operatorname{div}(\bar{H}_{R}\circledast K_{R})\\|_{L^{\infty}}$	$\displaystyle\leq C\\|\nabla K_{R}\\|_{L^{1}}\\|\nabla\bar{H}_{R}\\|_{L^{\infty}}\leq CR^{-8/3}.$

Since $0<R<1$ and $\beta>2/3$ ,

R^{-8/3}\leq R^{-2-\beta}.

Thus this term satisfies the bound asserted in (2.13). Because $H$ is only Lipschitz, two derivatives cannot be taken uniformly across the spherical interface before mollification. Instead, we place one derivative on the mollifier:

\|\nabla\operatorname{div}(F_{2}\ast\rho_{R^{\beta}})\|_{L^{\infty}}\leq C\|\operatorname{div}F_{2}\|_{L^{\infty}}\|\nabla\rho_{R^{\beta}}\|_{L^{1}}\leq CR^{-2-\beta}.

Together with the preceding estimates and Young’s inequality, this proves all asserted bounds for $\widetilde{P}_{2}$ and $\widetilde{F}_{2}$ .

The estimates for $\widetilde{H}_{R}$ follow from Lemma 2.3; for the second derivative, use

\|\nabla^{2}\widetilde{H}_{R}\|_{L^{q}}\leq\|\nabla\bar{H}_{R}\|_{L^{q}}\|\nabla\rho_{R^{\beta}}\|_{L^{1}}\leq CR^{\frac{2}{q}-\frac{5}{3}-\beta}.

Convolution preserves the integral. Finally, $\beta>2/3>\alpha$ implies $R^{\beta}<R^{\alpha}$ , so convolution enlarges each support by at most $R^{\beta}$ ; this yields the stated support in $\overline{B}_{3R^{\alpha}}$ . ∎

For $e\in\mathbb{S}^{2}$ , we use the rotated notation

\widetilde{H}_{R,e}(x):=Q_{e}\widetilde{H}_{R}(Q_{e}^{T}x),\qquad\widetilde{P}_{j,R,e}(x):=\widetilde{P}_{j}(Q_{e}^{T}x),

and

\widetilde{F}_{j,R,e}(x):=Q_{e}\widetilde{F}_{j}(Q_{e}^{T}x)Q_{e}^{T},\qquad j\in\{1,2\}.

Equations (2.11)–(2.12) and all the preceding estimates are rotation invariant, while

\frac{1}{R}\int_{\mathbb{R}^{3}}\widetilde{H}_{R,e}\,\mathrm{d}x=-2\pi e.

3 The moving Hill block

We now modulate the amplitude and scale of the localized Hill profile and translate its center along a time-dependent trajectory. The velocity of the center is chosen to match the intrinsic traveling speed of the profile; this cancels the leading transport term against the quadratic Euler term. Scale and amplitude modulation leave a distinguished vector source whose spatial mean is prescribed by (2.3). All remaining terms are placed in a symmetric stress. A Helmholtz correction restores incompressibility, and the final corollary incorporates viscosity.

Proposition 3.1 (Moving Hill block).

Let $I\subset\mathbb{R}$ be an interval, fix $e\in\mathbb{S}^{2}$ and $p\in(1,\infty)$ , and choose

\beta=\frac{14}{3}+\frac{3\alpha-2}{p}-6\alpha

as in Corollary 2.5. Shrinking $R_{0}$ if necessary, assume

3R_{0}^{\alpha}<\frac{1}{2}.

Let

\eta\in C_{c}^{\infty}(I),\qquad\mathfrak{R}\in C^{1}(\mathbb{T}^{3};(0,R_{0}]).

Choose $t_{0}\in I$ and $x_{0}\in\mathbb{T}^{3}$ , and let $X$ solve

\begin{cases}X^{\prime}(t)=-\dfrac{\eta(t)}{\mathfrak{R}(X(t))}e,\\ X(t_{0})=x_{0}.\end{cases}

(3.1)

Set $R(t):=\mathfrak{R}(X(t))$ .

Define the principal block

V^{p}(x,t):=\eta(t)\,\widetilde{H}_{R(t),e}(x-X(t)).

(3.2)

We suppress the parameters $R$ and $e$ on the pressure and error terms when no confusion can arise. The profile identities are

\frac{1}{R}\,e\cdot\nabla\widetilde{H}_{R,e}+\operatorname{div}(\widetilde{H}_{R,e}\otimes\widetilde{H}_{R,e})+\nabla\widetilde{P}_{1}=\operatorname{div}\widetilde{F}_{1},

\partial_{R}\widetilde{H}_{R,e}=\frac{1}{R}\widetilde{H}_{R,e}+\nabla\widetilde{P}_{2}+\operatorname{div}\widetilde{F}_{2}.

The support condition on $R_{0}$ allows the compactly supported Euclidean profiles to be periodized on $\mathbb{T}^{3}=(\mathbb{R}/\mathbb{Z})^{3}$ without overlap between distinct copies.

Since $V^{p}$ is periodic,

\int_{\mathbb{T}^{3}}\operatorname{div}V^{p}(x,t)\,\mathrm{d}x=0,

so the periodic inverse Laplacian below is well defined on $\operatorname{div}V^{p}$ . To restore incompressibility on $\mathbb{T}^{3}$ , define

V^{c}:=-\nabla\Delta^{-1}\operatorname{div}V^{p},\qquad V:=V^{p}+V^{c}.

Then

\int_{\mathbb{T}^{3}}V^{c}(x,t)\,\mathrm{d}x=0,\qquad\operatorname{div}V(\cdot,t)=0

for every $t\in I$ , and

\displaystyle\partial_{t}V+\operatorname{div}(V\otimes V)+\nabla P

\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}(\eta(t)R(t))\frac{1}{R(t)}\widetilde{H}_{R(t),e}(x-X(t))+\operatorname{div}F,

(3.3)

where $F$ is a symmetric matrix field. The following estimates hold:

The stress satisfies

\|F\|_{L_{t}^{\infty}L_{x}^{1}}\leq C\|\eta\|_{L^{\infty}}^{2}\|R^{\kappa}\|_{L^{\infty}(I)}(1+\|\nabla\log\mathfrak{R}\|_{L^{\infty}}),

where

\kappa=\min\left\{\frac{2+3\alpha}{2p}-4\alpha,2-6\alpha+\frac{3\alpha}{p},\frac{2}{p}-\frac{4}{3}\right\}.

For every $q\in(1,\infty)$ , the displayed spatial estimates below are understood uniformly in $t\in I$ ; all powers of the variable $R(t)$ are taken in $L^{\infty}(I)$ , and all unlabelled norms of $\mathfrak{R}$ are over $\mathbb{T}^{3}$ :

	$\displaystyle\\|V\\|_{L_{t}^{\infty}L_{x}^{q}}$	$\displaystyle\leq C\\|\eta\\|_{\infty}\\|R^{\frac{2}{q}-1}\\|_{L^{\infty}(I)},$
	$\displaystyle\\|\nabla V\\|_{L_{t}^{\infty}L_{x}^{q}}$	$\displaystyle\leq C\\|\eta\\|_{\infty}\\|R^{\frac{2}{q}-\frac{5}{3}}\\|_{L^{\infty}(I)},$
	$\displaystyle\\|\partial_{t}V\\|_{L_{t}^{\infty}L_{x}^{q}}$	$\displaystyle\leq C\\|\eta\\|_{\infty}^{2}\\|R^{-3}\\|_{L^{\infty}(I)}(1+\\|\nabla\mathfrak{R}\\|_{\infty})+C\\|\eta^{\prime}\\|_{\infty}\\|R^{-1}\\|_{L^{\infty}(I)},$
	$\displaystyle\\|\partial_{t}\nabla V\\|_{L_{t}^{\infty}L_{x}^{q}}$	$\displaystyle\leq C\\|\eta\\|_{\infty}^{2}\\|R^{-3-\beta}\\|_{L^{\infty}(I)}(1+\\|\nabla\mathfrak{R}\\|_{\infty})+C\\|\eta^{\prime}\\|_{\infty}\\|R^{-\frac{5}{3}}\\|_{L^{\infty}(I)}.$

3.

For every $q\in(1,\infty)$ , $\|R^{-1}\widetilde{H}_{R,e}\|_{L^{q}}\leq CR^{\frac{2}{q}-2}$ , and

$\int_{\mathbb{T}^{3}}\frac{1}{R}\widetilde{H}_{R,e}\,\mathrm{d}x=-2\pi e.$

In particular,

\int_{\mathbb{T}^{3}}V(x,t)\,\mathrm{d}x=-2\pi\eta(t)R(t)e.

Thus the block is not itself mean zero; the distinguished vector source in (3.3) is exactly the time derivative of its spatial mean.

Proof.

Write $Y:=x-X(t)$ . Differentiating (3.2) gives

	$\displaystyle\partial_{t}V^{p}(x,t)$	$\displaystyle=\eta^{\prime}(t)\widetilde{H}_{R(t),e}(Y)-\eta(t)X^{\prime}(t)\cdot\nabla\widetilde{H}_{R(t),e}(Y)$
		$\displaystyle\quad+\eta(t)R^{\prime}(t)\left[\partial_{R}\widetilde{H}_{R,e}\right]_{R=R(t)}(Y).$		(3.4)

Moreover,

\operatorname{div}(V^{p}\otimes V^{p})(x,t)=\eta(t)^{2}\,\operatorname{div}\bigl(\widetilde{H}_{R(t),e}\otimes\widetilde{H}_{R(t),e}\bigr)(Y).

By Corollary 2.5,

\operatorname{div}(\widetilde{H}_{R,e}\otimes\widetilde{H}_{R,e})+\nabla\widetilde{P}_{1}=-\frac{1}{R}(e\cdot\nabla)\widetilde{H}_{R,e}+\operatorname{div}\widetilde{F}_{1}.

The scale derivative satisfies

\partial_{R}\widetilde{H}_{R,e}=\frac{1}{R}\widetilde{H}_{R,e}+\nabla\widetilde{P}_{2}+\operatorname{div}\widetilde{F}_{2}.

Combining these identities with the trajectory equation (3.1), the transport term cancels and we obtain

\partial_{t}V^{p}+\operatorname{div}(V^{p}\otimes V^{p})+\nabla P^{p}=\frac{\mathrm{d}}{\mathrm{d}t}(\eta R)\frac{1}{R}\widetilde{H}_{R,e}(Y)+\operatorname{div}A,

(3.5)

where all profiles on the right are evaluated at $R=R(t)$ , and

P^{p}(x,t):=\eta(t)^{2}\widetilde{P}_{1,R(t),e}(Y)-\eta(t)R^{\prime}(t)\widetilde{P}_{2,R(t),e}(Y),

A(x,t):=\eta(t)^{2}\widetilde{F}_{1,R(t),e}(Y)+\eta(t)R^{\prime}(t)\widetilde{F}_{2,R(t),e}(Y).

Since $\partial_{t}V^{c}=-\nabla\partial_{t}\Delta^{-1}\operatorname{div}V^{p}$ , this term can be absorbed into the pressure. Let $\mathcal{R}$ denote the symmetric inverse-divergence operator on $\mathbb{T}^{3}$ introduced in [42]; we use its $L^{p}$ and order- $(-1)$ bounds in the form established in [23, 26]. The vector field $\operatorname{div}A$ has zero spatial mean, and hence

\operatorname{div}\mathcal{R}(\operatorname{div}A)=\operatorname{div}A.

Define

	$\displaystyle P$	$\displaystyle:=P^{p}+\partial_{t}\Delta^{-1}\operatorname{div}V^{p},$
	$\displaystyle F$	$\displaystyle:=\mathcal{R}(\operatorname{div}A)+V^{p}\otimes V^{c}+V^{c}\otimes V^{p}+V^{c}\otimes V^{c}.$		(3.6)

The tensor $F$ is symmetric. Adding the equation for $V^{c}$ to (3.5) now proves (3.3).

It remains to establish the estimates. The chain rule and (3.1) give

R^{\prime}(t)=\nabla\mathfrak{R}(X(t))\cdot X^{\prime}(t)=-\eta(t)e\cdot\nabla\log\mathfrak{R}(X(t)).

(3.7)

Set $F_{3}:=\mathcal{R}(\operatorname{div}A)$ . Since $\mathcal{R}\operatorname{div}$ is a Calderón–Zygmund operator on $\mathbb{T}^{3}$ [74], Corollary 2.5 and (3.7) give

	$\displaystyle\\|F_{3}\\|_{L_{t}^{\infty}L_{x}^{1}}$	$\displaystyle\leq C\\|F_{3}\\|_{L_{t}^{\infty}L_{x}^{p}}$
		$\displaystyle\leq C\\|\eta\\|_{L^{\infty}}^{2}\left(\\|R^{2-6\alpha+\frac{3\alpha}{p}}\\|_{L^{\infty}(I)}+\\|\nabla\log\mathfrak{R}\\|_{L^{\infty}}\\|R^{\frac{2}{p}-\frac{4}{3}}\\|_{L^{\infty}(I)}\right).$		(3.8)

We use two complementary bounds for the Helmholtz correction. For every $s\in(1,\infty)$ , Calderón–Zygmund boundedness gives

\|V^{c}\|_{L^{s}}\leq C\|V^{p}\|_{L^{s}},

while Poincaré’s inequality and the definition of $V^{c}$ give

\|V^{c}\|_{L^{s}}\leq C\|\nabla V^{c}\|_{L^{s}}\leq C\|\operatorname{div}V^{p}\|_{L^{s}}.

In particular, at $s=2p$ ,

\|V^{p}\|_{L^{2p}}\leq C\|\eta\|_{L^{\infty}}R^{\frac{1}{p}-1},\qquad\|V^{c}\|_{L^{2p}}\leq C\|\eta\|_{L^{\infty}}R^{1-4\alpha+\frac{3\alpha}{2p}}.

The first of the two correction bounds also implies $\|V^{c}\otimes V^{c}\|_{L^{p}}\leq C\|V^{p}\|_{L^{2p}}\|V^{c}\|_{L^{2p}}$ . Therefore,

\|V^{p}\otimes V^{c}+V^{c}\otimes V^{p}+V^{c}\otimes V^{c}\|_{L^{p}}\leq C\|\eta\|_{L^{\infty}}^{2}R^{\frac{2+3\alpha}{2p}-4\alpha}.

(3.9)

Combining (3.8), (3.9), and (3.6), and using $|\mathbb{T}^{3}|=1$ , yields the asserted stress estimate with

\kappa=\min\left\{\frac{2+3\alpha}{2p}-4\alpha,\,2-6\alpha+\frac{3\alpha}{p},\,\frac{2}{p}-\frac{4}{3}\right\}.

For the remaining estimates, fix $q\in(1,\infty)$ . The profile estimates in Corollary 2.5 give

\|V^{p}\|_{L^{q}}\leq C|\eta|R^{\frac{2}{q}-1},\qquad\|\nabla V^{p}\|_{L^{q}}\leq C|\eta|R^{\frac{2}{q}-\frac{5}{3}}.

The Calderón–Zygmund bounds for the Helmholtz projection give

	$\displaystyle\\|V^{c}\\|_{L^{q}}$	$\displaystyle\leq C\\|V^{p}\\|_{L^{q}},$	$\displaystyle\\|\nabla V^{c}\\|_{L^{q}}$	$\displaystyle\leq C\\|\nabla V^{p}\\|_{L^{q}},$
	$\displaystyle\\|\partial_{t}V^{c}\\|_{L^{q}}$	$\displaystyle\leq C\\|\partial_{t}V^{p}\\|_{L^{q}},$	$\displaystyle\\|\partial_{t}\nabla V^{c}\\|_{L^{q}}$	$\displaystyle\leq C\\|\partial_{t}\nabla V^{p}\\|_{L^{q}}.$

It remains to estimate the two time derivatives of $V^{p}$ . Equation (3.4) and Corollary 2.5 yield, at each fixed $t$ ,

	$\displaystyle\\|\partial_{t}V^{p}\\|_{L^{\infty}}$	$\displaystyle\leq\frac{\|\eta\|^{2}}{R}\\|\nabla\widetilde{H}_{R,e}\\|_{L^{\infty}}+\|\eta^{\prime}\|\\|\widetilde{H}_{R,e}\\|_{L^{\infty}}$
		$\displaystyle\quad+\|\eta R^{\prime}\|\left(\frac{1}{R}\\|\widetilde{H}_{R,e}\\|_{L^{\infty}}+\\|\nabla\widetilde{P}_{2}\\|_{L^{\infty}}+\\|\operatorname{div}\widetilde{F}_{2}\\|_{L^{\infty}}\right)$
		$\displaystyle\leq C\left(\|\eta\|^{2}R^{-\frac{8}{3}}+\|\eta^{\prime}\|R^{-1}+\|\eta R^{\prime}\|R^{-2}\right),$

and

	$\displaystyle\\|\partial_{t}\nabla V^{p}\\|_{L^{\infty}}$	$\displaystyle\leq\frac{\|\eta\|^{2}}{R}\\|\nabla^{2}\widetilde{H}_{R,e}\\|_{L^{\infty}}+\|\eta^{\prime}\|\\|\nabla\widetilde{H}_{R,e}\\|_{L^{\infty}}$
		$\displaystyle\quad+\|\eta R^{\prime}\|\left(\frac{1}{R}\\|\nabla\widetilde{H}_{R,e}\\|_{L^{\infty}}+\\|\nabla^{2}\widetilde{P}_{2}\\|_{L^{\infty}}+\\|\nabla\operatorname{div}\widetilde{F}_{2}\\|_{L^{\infty}}\right)$
		$\displaystyle\leq C\left(\|\eta\|^{2}R^{-\frac{8}{3}-\beta}+\|\eta^{\prime}\|R^{-\frac{5}{3}}+\|\eta R^{\prime}\|R^{-2-\beta}\right).$

In the last two estimates we used $\alpha<2/3$ and $\beta>2/3$ to dominate the pressure terms by the displayed powers of $R$ . Using (3.7) in the form

|R^{\prime}(t)|\leq|\eta(t)|R(t)^{-1}\|\nabla\mathfrak{R}\|_{L^{\infty}},

together with $R\leq R_{0}<1$ , gives the asserted time-derivative bounds. The $L^{q}$ -estimates follow from the $L^{\infty}$ -bounds because $|\mathbb{T}^{3}|=1$ .

The final assertion follows from Corollary 2.5 and is unchanged by periodization. ∎

Corollary 3.2 (Navier–Stokes moving Hill block).

Under the assumptions of Proposition 3.1, the divergence-free block $V$ satisfies

\partial_{t}V+\operatorname{div}(V\otimes V)-\nu\Delta V+\nabla P=\frac{\mathrm{d}}{\mathrm{d}t}(\eta(t)R(t))\frac{1}{R(t)}\widetilde{H}_{R(t),e}(x-X(t))+\operatorname{div}F^{NS},

(3.10)

where $F^{NS}$ is symmetric and satisfies

\|F^{NS}\|_{L_{t}^{\infty}L_{x}^{1}}\leq C\|\eta\|_{L^{\infty}}^{2}\|R^{\kappa}\|_{L^{\infty}(I)}(1+\|\nabla\log\mathfrak{R}\|_{L^{\infty}})+C\nu\|\eta\|_{L^{\infty}}\|R^{\frac{2}{p}-\frac{5}{3}}\|_{L^{\infty}(I)}.

Proof.

Since $\operatorname{div}V=0$ ,

-\nu\Delta V=\operatorname{div}\!\left[-\nu\bigl(\nabla V+(\nabla V)^{T}\bigr)\right].

Thus, treating viscosity as part of the stress gives

	$\displaystyle\partial_{t}V+\operatorname{div}(V\otimes V)-\nu\Delta V+\nabla P$	$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}t}(\eta(t)R(t))\frac{1}{R(t)}\widetilde{H}_{R(t),e}(x-X(t))$
		$\displaystyle\quad+\operatorname{div}\!\left(F-\nu(\nabla V+(\nabla V)^{T})\right).$

Thus (3.10) follows after setting

F^{NS}:=F-\nu(\nabla V+(\nabla V)^{T}).

This tensor is symmetric. By Proposition 3.1,

\|\nu(\nabla V+(\nabla V)^{T})\|_{L_{t}^{\infty}L_{x}^{1}}\leq C\nu\|\nabla V\|_{L_{t}^{\infty}L_{x}^{p}}\leq C\nu\|\eta\|_{L^{\infty}}\|R^{\frac{2}{p}-\frac{5}{3}}\|_{L^{\infty}(I)}.

∎

4 The iteration step

We carry out one stage of the Navier–Stokes–Reynolds iteration. Starting from an admissible triple $(u_{q},p_{q},\mathcal{E}_{q})$ , we regularize the background flow, decompose the mollified stress into rank-one components, and assign each component to a moving Hill block on a short time interval. An auxiliary source and a temporal corrector then convert the orbit-averaged cancellation into a pointwise-in-time equation. Here admissible means that the velocity is continuous in time and the triple is smooth in the interior of each time cell, with bounded one-sided derivatives at the cell interfaces; the Navier–Stokes–Reynolds system is understood distributionally. This class is preserved by the construction below and becomes smooth after the mollification at the beginning of the next stage.

To avoid boundary artifacts in the space–time convolution, all triples are constructed on a fixed open interval $I_{\ast}\supset[0,1]$ , and the iteration is restricted to $[0,1]$ only after mollification. All time-dependent estimates below are understood on $I_{\ast}$ ; this harmless convention will not be repeated.

We begin by fixing the parameter hierarchy. At the $q$ th stage, $\lambda_{q}$ controls the density of the rational orbits, $\delta_{q}$ measures the Reynolds stress, $r_{q}$ is the base concentration scale, and $\tau_{q}$ is the length of a time cell. For $q\in\mathbb{Z}_{\geq 0}$ , set

\lambda_{q+1}=\lambda_{q}^{\sigma},\quad\delta_{q}=\lambda_{1}^{2\gamma}\lambda_{q}^{-\gamma},\quad r_{q+1}=\lambda_{q+1}^{-\mu},\quad\tau_{q+1}=\lambda_{q+1}^{-\eta}.

The constants $\mu,\eta,\gamma,\sigma$ are positive, while $\lambda_{0}$ depends on the initial data and will be chosen sufficiently large. We also take $\lambda_{0}$ , $\sigma$ , and $\eta$ to be integers. For the intrinsic moving-block estimates we fix, throughout this section,

p_{0}=\frac{28}{27},\qquad\alpha=\frac{2}{11},\qquad\beta=\frac{502}{231},\qquad\kappa=\frac{1}{2}.

(4.1)

The numerical choices of the remaining parameters, and the verification of all exponent inequalities, are given in the parameter-selection subsection 5.3 below.

Suppose that the Navier–Stokes–Reynolds system at the $q$ th stage is

	$\displaystyle\partial_{t}u_{q}+\operatorname{div}(u_{q}\otimes u_{q})-\nu\Delta u_{q}+\nabla p_{q}$	$\displaystyle=\operatorname{div}(\mathcal{E}_{q}),$		(4.2)
	$\displaystyle\operatorname{div}u_{q}$	$\displaystyle=0.$

The triple is admissible on $\mathbb{T}^{3}\times I_{\ast}$ , and the induction assumptions are

\|\mathcal{E}_{q}\|_{L_{t}^{\infty}L_{x}^{1}}\leq\delta_{q+1},\quad\|u_{q}\|_{L_{t}^{\infty}L_{x}^{2}}\leq 2\delta_{0}^{\frac{1}{2}}-\delta_{q}^{\frac{1}{2}},\quad\|(\nabla_{x}u_{q},\partial_{t}u_{q})\|_{L_{t}^{\infty}L_{x}^{4}}\leq\lambda_{q}^{n}.

(4.3)

Here $n>0$ is fixed in the parameter selection below.

Proposition 4.1.

Let $\bar{p}=\frac{6}{5}+5\times 10^{-5}$ . Choose the constants $\mu,\eta,\gamma,\sigma$ as in the parameter selection below, and let $\lambda_{0}$ be sufficiently large. If $(u_{q},p_{q},\mathcal{E}_{q})$ is admissible and satisfies (4.2) and (4.3), then there exists an admissible triple $(u_{q+1},p_{q+1},\mathcal{E}_{q+1})$ satisfying (4.2) and the following bounds:

1.

$\|\mathcal{E}_{q+1}\|_{L_{t}^{\infty}L_{x}^{1}}\leq\delta_{q+2}$ , $\|u_{q+1}\|_{L_{t}^{\infty}L_{x}^{2}}\leq 2\delta_{0}^{1/2}-\delta_{q+1}^{1/2}$ , and $\|(\nabla_{x}u_{q+1},\partial_{t}u_{q+1})\|_{L_{t}^{\infty}L_{x}^{4}}\leq\lambda_{q+1}^{n}$ ,
2.

$\|u_{q+1}-u_{q}\|_{L_{t}^{\infty}L_{x}^{2}}\leq C\delta_{q+1}^{1/2}$ ,

There exist $\alpha_{0},\vartheta>0$ , independent of $q$ , such that

\|\nabla u_{q+1}\|_{C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}}\leq\|\nabla u_{q}\|_{C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}}+C\delta_{q+1}^{\vartheta},

4.

$\|u_{q+1}(\cdot,0)-u_{q}(\cdot,0)\|_{L^{2}}\leq C\lambda_{q}^{-1/5}$ and $\|u_{q+1}(\cdot,1)-u_{q}(\cdot,1)\|_{L^{2}}\leq C\lambda_{q}^{-1/5}$ .

4.1 Mollification step

We regularize the background flow and Reynolds stress at a space–time scale $l$ . This provides the derivative bounds needed to define the space-dependent radii and trajectories of the vortex cores. The commutator generated by mollifying the nonlinear term is included in the new Reynolds stress. Fix a sufficiently small absolute constant $c_{0}>0$ and set

l:=c_{0}\lambda_{q+1}^{-n/\sigma-\gamma}.

Let $\rho_{l}(t,x)=l^{-4}\rho(t/l,x/l)$ be a smooth space–time mollifier on $\mathbb{R}\times\mathbb{T}^{3}$ , where $\rho$ is supported in the unit space–time ball, and set

u_{l}=\rho_{l}\ast u_{q},\quad p_{l}=\rho_{l}\ast p_{q},\quad\mathcal{E}_{l}=\rho_{l}\ast\mathcal{E}_{q}+u_{l}\otimes u_{l}-(u_{q}\otimes u_{q})\ast\rho_{l}.

(4.4)

Then $(u_{l},p_{l},\mathcal{E}_{l})$ solves (4.2) and satisfies

	$\displaystyle\\|\mathcal{E}_{l}\\|_{L_{t}^{\infty}L_{x}^{1}}$	$\displaystyle\leq\\|\rho_{l}\ast\mathcal{E}_{q}\\|_{L_{t}^{\infty}L_{x}^{1}}+\\|u_{l}\otimes u_{l}-(u_{q}\otimes u_{q})\ast\rho_{l}\\|_{L_{t}^{\infty}L_{x}^{1}}$
		$\displaystyle\leq\delta_{q+1}+Cl\\|u_{q}\\|_{L_{t}^{\infty}L_{x}^{2}}\\|(\nabla_{x}u_{q},\partial_{t}u_{q})\\|_{L_{t}^{\infty}L_{x}^{2}}$
		$\displaystyle\leq\delta_{q+1}+Cc_{0}\delta_{0}^{\frac{1}{2}}\lambda_{q+1}^{-\gamma}$
		$\displaystyle\leq 2\delta_{q+1}.$

Here the last inequality follows from $\delta_{0}^{1/2}\leq\lambda_{1}^{2\gamma}$ and the definition of $\delta_{q+1}$ . This verifies that the smaller scale still controls the mollification commutator.

We also check the inverse powers of $l$ . At $q=0$ the initial triple is fixed and smooth before $\lambda_{0}$ is chosen, so the following bounds hold after increasing $\lambda_{0}$ . For $q\geq 1$ , we use $\lambda_{q+1}\geq\lambda_{2}=\lambda_{1}^{\sigma}$ and $W^{1,4}(\mathbb{T}^{3})\hookrightarrow L^{\infty}(\mathbb{T}^{3})$ to obtain

	$\displaystyle\\|\mathcal{E}_{l}\\|_{C^{0}_{x,t}}$	$\displaystyle\leq Cl^{-3}\\|\mathcal{E}_{q}\\|_{L_{t}^{\infty}L_{x}^{1}}+C\lambda_{q}^{2n}$
		$\displaystyle\leq C\lambda_{1}^{2\gamma}\lambda_{q+1}^{3n/\sigma+2\gamma}+C\lambda_{q+1}^{2n/\sigma}\leq C\lambda_{q+1}^{3n/\sigma+3\gamma},$
	$\displaystyle\\|\mathcal{E}_{l}\\|_{\dot{C}^{1}_{x,t}}$	$\displaystyle\leq Cl^{-4}\\|\mathcal{E}_{q}\\|_{L_{t}^{\infty}L_{x}^{1}}+Cl^{-1}\lambda_{q}^{2n}$
		$\displaystyle\leq C\lambda_{1}^{2\gamma}\lambda_{q+1}^{4n/\sigma+3\gamma}+C\lambda_{q+1}^{3n/\sigma+\gamma}\leq C\lambda_{q+1}^{4n/\sigma+4\gamma}.$

In the last inequalities we used $\lambda_{1}^{2\gamma}\leq\lambda_{q+1}^{2\gamma/\sigma}$ . Thus all subsequent bounds involving powers of $l$ retain the exponents used below.

4.2 Error decomposition

The decomposition of a symmetric matrix into nine positive rank-one components is the geometric lemma of [83, 42, 23]. The refinement producing directions with rational component ratios and orbit periods comparable to $\lambda_{q+1}^{2}$ follows [14]; the perturbation matrix $K$ and the density argument in item (i) below are specific to the present three-dimensional setting.

We resolve the mollified Reynolds stress, up to a scalar pressure, into finitely many positive rank-one components. The directions are chosen to have rational components, periods comparable to $\lambda_{q+1}^{2}$ , and $O(\lambda_{q+1}^{-1})$ -dense trajectories; they will later determine the trajectories of the moving vortex cores. The positive coefficients determine both the amplitudes and the spatial scales of the corresponding perturbations.

Lemma 4.2 (Error Decomposition).

Given $\lambda_{q+1}\geq 192$ and $\delta_{q+1}>0$ , there exist unit vectors $\xi_{i}\in\mathbb{R}^{3}$ , $i=1,\dots,9$ , such that, for every $\mathcal{E}_{l}\in C^{\infty}(\mathbb{T}^{3}\times I_{\ast};\operatorname{Sym}_{3})$ , the following statements hold.

(i)

The curve $s\mapsto s\xi_{i}$ on $\mathbb{T}^{3}$ has minimal period $c_{i}\lambda_{q+1}^{2}$ for some $c_{i}\in[1/4,2]$ , and its image is $C_{0}\lambda_{q+1}^{-1}$ -dense in $\mathbb{T}^{3}$ , where $C_{0}$ is independent of $q$ and $i$ .

(ii)

The following decomposition holds:

\displaystyle-\operatorname{div}(\mathcal{E}_{l})=\operatorname{div}\left(\sum_{i=1}^{9}a_{i}(x,t)\xi_{i}\otimes\xi_{i}\right)-\nabla\zeta.

(4.5)

(iii)

The functions $a_{i}(x,t)$ are smooth and satisfy

$\displaystyle a_{i}(x,t)$	$\displaystyle\geq\delta_{q+1},$	(4.6)
$\displaystyle\\|a_{i}\\|_{L_{t}^{\infty}L_{x}^{1}}$	$\displaystyle\leq 192\delta_{q+1},$
$\displaystyle\\|a_{i}\\|_{C^{0}_{x,t}}$	$\displaystyle\leq C\bigl(\delta_{q+1}+\\|\mathcal{E}_{l}\\|_{C^{0}_{x,t}}\bigr),$
$\displaystyle\\|a_{i}\\|_{C^{1}_{x,t}}$	$\displaystyle\leq C\bigl(\delta_{q+1}+\\|\mathcal{E}_{l}\\|_{C^{1}_{x,t}}\bigr).$

Proof.

Define the following nine unit vectors:

$\displaystyle e_{1}$	$\displaystyle=(1,0,0)^{T},$	$\displaystyle e_{2}$	$\displaystyle=(0,1,0)^{T},$	$\displaystyle e_{3}$	$\displaystyle=(0,0,1)^{T},$
$\displaystyle e_{4}$	$\displaystyle=2^{-1/2}(1,1,0)^{T},$	$\displaystyle e_{5}$	$\displaystyle=2^{-1/2}(0,1,1)^{T},$	$\displaystyle e_{6}$	$\displaystyle=2^{-1/2}(1,0,1)^{T},$
$\displaystyle e_{7}$	$\displaystyle=2^{-1/2}(1,-1,0)^{T},$	$\displaystyle e_{8}$	$\displaystyle=2^{-1/2}(0,1,-1)^{T},$	$\displaystyle e_{9}$	$\displaystyle=2^{-1/2}(1,0,-1)^{T}.$

For a symmetric $3\times 3$ matrix $\bar{\mathcal{E}}$ , use the decomposition

\displaystyle\bar{\mathcal{E}}=\sum_{i=1}^{9}\bar{\Gamma}_{i}(\bar{\mathcal{E}})\,e_{i}\otimes e_{i},

(4.7)

where $\bar{\Gamma}_{i}$ are smooth functions given by

$\displaystyle\bar{\Gamma}_{1}(\bar{\mathcal{E}})$	$\displaystyle:=\bar{\mathcal{E}}_{11}-\bar{\mathcal{E}}_{12}-\bar{\mathcal{E}}_{13}-\frac{1}{2},$
$\displaystyle\bar{\Gamma}_{2}(\bar{\mathcal{E}})$	$\displaystyle:=\bar{\mathcal{E}}_{22}-\bar{\mathcal{E}}_{12}-\bar{\mathcal{E}}_{23}-\frac{1}{2},$
$\displaystyle\bar{\Gamma}_{3}(\bar{\mathcal{E}})$	$\displaystyle:=\bar{\mathcal{E}}_{33}-\bar{\mathcal{E}}_{23}-\bar{\mathcal{E}}_{13}-\frac{1}{2},$
$\displaystyle\bar{\Gamma}_{4}(\bar{\mathcal{E}})$	$\displaystyle:=2\bar{\mathcal{E}}_{12}+\frac{1}{4},$	$\displaystyle\bar{\Gamma}_{5}(\bar{\mathcal{E}})$	$\displaystyle:=2\bar{\mathcal{E}}_{23}+\frac{1}{4},$
$\displaystyle\bar{\Gamma}_{6}(\bar{\mathcal{E}})$	$\displaystyle:=2\bar{\mathcal{E}}_{13}+\frac{1}{4},$	$\displaystyle\bar{\Gamma}_{7}(\bar{\mathcal{E}})$	$\displaystyle:=\frac{1}{4},$
$\displaystyle\bar{\Gamma}_{8}(\bar{\mathcal{E}})$	$\displaystyle:=\frac{1}{4},$	$\displaystyle\bar{\Gamma}_{9}(\bar{\mathcal{E}})$	$\displaystyle:=\frac{1}{4}.$

If $\|\bar{\mathcal{E}}-I_{3\times 3}\|_{\infty}<1/16$ , then $1/8\leq\bar{\Gamma}_{i}(\bar{\mathcal{E}})\leq 2$ , and $\max_{i,j,k}\left|\frac{\partial\bar{\Gamma}_{i}}{\partial\bar{\mathcal{E}}_{j,k}}\right|\leq 2$ .

Put $N=\lambda_{q+1}$ , and let $K$ denote the matrix

K=\frac{1}{N^{2}}\begin{pmatrix}N^{2}&1&N\\ N&N^{2}+1&1\\ 1&N&N^{2}+2\end{pmatrix}.

Define vectors $\xi_{1},\dots,\xi_{9}\in\mathbb{R}^{3}$ , whose component ratios are rational, by

\displaystyle\xi_{i}\coloneqq-\frac{Ke_{i}}{|Ke_{i}|},\qquad i=1,\dots,9.

To verify item (i), write

v_{i}=e_{i}\quad(i=1,2,3),\qquad v_{i}=\sqrt{2}\,e_{i}\quad(i=4,\ldots,9),

so that $v_{i}\in\mathbb{Z}^{3}$ , and set

M=N^{2}K=N^{2}I+NA+B,\qquad m_{i}=Mv_{i},

where

A=\begin{pmatrix}0&0&1\\ 1&0&0\\ 0&1&0\end{pmatrix},\qquad B=\begin{pmatrix}0&1&0\\ 0&1&1\\ 1&0&2\end{pmatrix}.

Then $Ke_{i}$ is parallel to $m_{i}$ , and hence $\xi_{i}=-m_{i}/|m_{i}|$ . For example,

m_{1}=(N^{2},N,1),\quad m_{4}=(N^{2}+1,N^{2}+N+1,N+1),\quad m_{6}=(N^{2}+N,N+1,N^{2}+3).

The vectors $m_{i}$ need not themselves be primitive. If $d_{i}$ denotes the greatest common divisor of their components, a direct calculation gives

d_{i}=1\quad(i\neq 6,9),\qquad d_{6}=\gcd(N+1,4),\qquad d_{9}=\gcd(N-1,2).

Thus $1\leq d_{i}\leq 4$ , and $\widehat{m}_{i}:=m_{i}/d_{i}$ has relatively prime components. In particular,

\xi_{i}=-\frac{\widehat{m}_{i}}{|\widehat{m}_{i}|}=-\frac{m_{i}}{|m_{i}|}.

Since $A$ is orthogonal, $\|B\|\leq\sqrt{8}$ , and $|v_{i}|\leq\sqrt{2}$ , the columns of $M$ and the reverse triangle inequality give, for $N\geq 192$ ,

N^{2}\leq|m_{i}|\leq 2N^{2}.

Since $\widehat{m}_{i}$ is primitive, the minimal period is

|\widehat{m}_{i}|=c_{i}N^{2}.

As $1\leq d_{i}\leq 4$ and $N^{2}\leq|m_{i}|\leq 2N^{2}$ , we have

\frac{1}{4}\leq c_{i}=\frac{|m_{i}|}{d_{i}N^{2}}\leq 2.

This proves the period assertion in item (i) without any parity assumption on $N$ .

It remains to verify the density assertion. Notice first that $m_{i}$ and $\widehat{m}_{i}$ have the same annihilator lattice. A direct computation gives

\bigl(\det(v_{i},Av_{i},Bv_{i})\bigr)_{i=1}^{9}=(1,1,1,1,1,4,-1,-1,-2),

so $v_{i},Av_{i},Bv_{i}$ span $\mathbb{R}^{3}$ for every $i$ . If $n\in\mathbb{Z}^{3}$ satisfies $n\cdot m_{i}=0$ , then

N^{2}(n\cdot v_{i})+N(n\cdot Av_{i})+n\cdot Bv_{i}=0.

The three scalar products are integers. If $0<|n|<N/4$ and $n\cdot v_{i}\neq 0$ , then

N^{2}\leq N\sqrt{2}\,|n|+4|n|<\frac{\sqrt{2}}{4}N^{2}+N<N^{2},

a contradiction. Hence $n\cdot v_{i}=0$ . If $n\cdot Av_{i}\neq 0$ , then $N\leq|n\cdot Bv_{i}|\leq 4|n|<N$ , again a contradiction. Thus $n$ is orthogonal to $v_{i},Av_{i},Bv_{i}$ , and hence $n=0$ . Consequently,

\min\{|n|:n\in\mathbb{Z}^{3}\setminus\{0\},\ n\cdot m_{i}=0\}\geq\frac{N}{4}.

Let $\pi_{i}$ be the orthogonal projection onto $m_{i}^{\perp}$ . The dual of the two-dimensional lattice $\pi_{i}(\mathbb{Z}^{3})$ is $\{n\in\mathbb{Z}^{3}:n\cdot m_{i}=0\}$ . The standard two-dimensional transference estimate for the covering radius therefore gives

\sup_{x\in\mathbb{T}^{3}}\operatorname{dist}\bigl(x,\{s\xi_{i}:s\in\mathbb{R}\}\bigr)\leq\frac{C_{0}}{N}.

For a symmetric $3\times 3$ matrix $\mathcal{E}$ , apply (4.7) to $\bar{\mathcal{E}}=K^{-1}\mathcal{E}K^{-T}$ and multiply on the left by $K$ and on the right by $K^{T}$ . This gives

\displaystyle\mathcal{E}=\sum_{i=1}^{9}\Gamma_{i}(\mathcal{E})\,\xi_{i}\otimes\xi_{i},\qquad\mbox{where }\Gamma_{i}(\mathcal{E})\coloneqq|Ke_{i}|^{2}\,\bar{\Gamma}_{i}(K^{-1}\mathcal{E}K^{-T}),

\displaystyle\max_{i,j,k}\left|\frac{\partial\Gamma_{i}}{\partial\mathcal{E}_{j,k}}\right|\leq 4.

(4.8)

Since $K=I+O(N^{-1})$ , the condition $N\geq 192$ and the choice

\zeta=32\bigl(|\mathcal{E}_{l}|^{2}+\delta_{q+1}^{2}\bigr)^{1/2}

ensure that

\left\|K^{-1}\left(I-\frac{\mathcal{E}_{l}}{\zeta}\right)K^{-T}-I\right\|_{\infty}<\frac{1}{16}.

Thus the argument of every $\Gamma_{i}$ used below lies in the positivity neighborhood of the geometric decomposition.

Next, we define

\displaystyle a_{i}(x,t)\coloneqq\zeta(x,t)\,\Gamma_{i}\left(I_{3\times 3}-\frac{1}{\zeta(x,t)}\mathcal{E}_{l}(x,t)\right),\qquad\text{where}\quad\zeta(x,t)\coloneqq 32\left(|\mathcal{E}_{l}(x,t)|^{2}+\delta_{q+1}^{2}\right)^{\frac{1}{2}}.

It follows that

\displaystyle\sum a_{i}(x,t)\xi_{i}\otimes\xi_{i}=-\mathcal{E}_{l}(x,t)+\zeta(x,t)I_{3\times 3}.

Thus item (ii) holds. The bounds $1/8\leq\Gamma_{i}\leq 2$ , together with $\zeta\geq 32\delta_{q+1}$ , give $a_{i}\geq 4\delta_{q+1}$ , and hence the stated lower bound. Moreover, $\zeta\leq 32(|\mathcal{E}_{l}|+\delta_{q+1})$ ; using $\|\mathcal{E}_{l}\|_{L_{t}^{\infty}L_{x}^{1}}\leq 2\delta_{q+1}$ gives the claimed $L_{t}^{\infty}L_{x}^{1}$ -estimate. Finally, differentiating the formula for $a_{i}$ , using $|\nabla_{x,t}\zeta|\leq 32|\nabla_{x,t}\mathcal{E}_{l}|$ , and applying (4.8) gives

\displaystyle\|a_{i}\|_{C^{0}_{x,t}}\leq C(\delta_{q+1}+\|\mathcal{E}_{l}\|_{C^{0}_{x,t}}),\qquad\|a_{i}\|_{C^{1}_{x,t}}\leq C\bigl(\delta_{q+1}+\|\mathcal{E}_{l}\|_{C^{1}_{x,t}}\bigr).

∎

4.3 Time partition and cutoff functions

The nine directions from Lemma 4.2 are activated successively, thereby avoiding interactions between distinct directional families. We partition time into cells and divide each cell into nine subintervals, one for each direction. On every cell the coefficients are frozen at their temporal averages; the resulting discrepancy will later be included in the temporal error. Smooth cutoffs make the perturbations vanish near the endpoints of their active intervals.

Because $\tau_{q+1}^{-1}\in\mathbb{N}$ , set $\mathcal{K}_{q+1}:=\mathbb{Z}$ and partition the time axis into cells

\displaystyle\mathcal{T}^{k}\coloneqq[k\tau_{q+1},(k+1)\tau_{q+1}),\qquad k\in\mathcal{K}_{q+1}.

Only the finitely many cells meeting $I_{\ast}$ enter the construction on that interval. The integrality assumption makes both $0$ and $1$ cell interfaces.

Each interval $\mathcal{T}^{k}$ is further divided into nine subintervals of equal length:

\mathcal{T}^{k}_{i}\coloneqq\left[\tau_{q+1}\left(k+\frac{i-1}{9}\right),\;\tau_{q+1}\left(k+\frac{i}{9}\right)\right)\,,\quad i=1,\dots 9\,,\quad k\in\mathcal{K}_{q+1}\,.

Thus $\mathcal{T}^{k}=\bigcup_{i=1}^{9}\mathcal{T}^{k}_{i}$ . For the cutoff construction, we also use the slightly shorter interval

\displaystyle\bar{\mathcal{T}}^{k}_{i}\coloneqq\left[\tau_{q+1}\left(k+\frac{i-1}{9}+\frac{1}{\lambda_{q+1}}\right),\;\tau_{q+1}\left(k+\frac{i}{9}-\frac{1}{\lambda_{q+1}}\right)\right)\,,\quad i=1,2,\dots,9\,,\quad k\in\mathcal{K}_{q+1}\,.

(4.9)

We use the coefficients $a_{i}(x,t)$ from Lemma 4.2 and denote their time averages by

a_{i}^{k}(x):=\frac{1}{\tau_{q+1}}\int_{\mathcal{T}^{k}}a_{i}(x,t)\,\mathrm{d}t.

Equation (4.6) gives

\displaystyle a_{i}^{k}(x)\geq\delta_{q+1}\,,\quad\|a_{i}^{k}\|_{L^{1}_{x}}\leq 192\delta_{q+1}\,,\quad\|a_{i}^{k}\|_{C^{0}_{x}}\leq C\lambda^{3\frac{n}{\sigma}+3\gamma}_{q+1}\,,\quad\|a_{i}^{k}\|_{C^{1}_{x}}\leq C\lambda^{4\frac{n}{\sigma}+4\gamma}_{q+1}\,.

(4.10)

Moreover,

\|a_{i}^{k}-a_{i}\|_{L_{t}^{\infty}(\mathcal{T}^{k};L_{x}^{1})}\leq\tau_{q+1}\|\partial_{t}a_{i}\|_{L_{t}^{\infty}(\mathcal{T}^{k};L_{x}^{1})}\leq\tau_{q+1}\lambda^{4\frac{n}{\sigma}+4\gamma}_{q+1}.

Thus the freezing error is small when $\tau_{q+1}$ is sufficiently small.

For $k\in\mathcal{K}_{q+1}$ and $i\in\{1,\dots,9\}$ , choose a smooth cutoff $\zeta_{i}^{k}:\mathbb{R}\to[0,1]$ such that $\operatorname{supp}\zeta_{i}^{k}\subset\operatorname{int}(\mathcal{T}_{i}^{k})$ , $\zeta_{i}^{k}\equiv 1$ on $\bar{\mathcal{T}}_{i}^{k}$ , and

\|\frac{\mathrm{d}}{\mathrm{d}t}\zeta_{i}^{k}\|_{L^{\infty}_{t}}\leq 10\frac{\lambda_{q+1}}{\tau_{q+1}}.

(4.11)

4.4 Space-dependent vortex scale

We couple the radius of each core to the local size of the corresponding frozen stress coefficient. This choice produces the weighted orbit average needed to recover the tensor $a_{i}^{k}\xi_{i}\otimes\xi_{i}$ , while $r_{q+1}$ determines the overall concentration scale. The separation condition below ensures that the localized cores remain narrower than the spacing of the rational orbits. We set

\displaystyle\mathfrak{R}^{k}_{i}(x)\coloneqq r_{q+1}a^{k}_{i}(x)\,.

(4.12)

Because $\operatorname{supp}\widetilde{H}_{R,e}\subset B_{3R^{\alpha}}$ , we choose $r_{q+1}$ small enough that

6\sup_{x\in\mathbb{T}^{3}}(\mathfrak{R}_{i}^{k}(x))^{\alpha}\leq\lambda_{q+1}^{-1},\qquad i=1,\dots,9,\quad k\in\mathcal{K}_{q+1}.

(4.13)

Thus the diameter of every principal core is at most the geometric spacing scale $\lambda_{q+1}^{-1}$ . In particular, the Hill source and the auxiliary source introduced below are contained, after a common translation, in a ball of radius $\lambda_{q+1}^{-1}$ .

4.5 Trajectory of the vortex core

We next choose the amplitude and trajectory of each core. The amplitude is normalized by the spatial mean of $a_{i}^{k}$ , while the center moves at the intrinsic Hill speed in the direction $\xi_{i}$ . The initial point is selected so that the average of $a_{i}^{k}$ along one rational orbit agrees with its torus average. This normalization produces the desired directional stress after time averaging.

The cutoff used in Proposition 3.1 is the scalar multiple of $\zeta_{i}^{k}$ given by

\displaystyle\eta(t)=\eta^{k}_{i}\zeta^{k}_{i}(t),

where $\eta_{i}^{k}$ is chosen to produce exact cancellation:

\displaystyle(\eta_{i}^{k})^{2}=\frac{9}{2\pi}\int_{\mathbb{T}^{3}}a_{i}^{k}(x)\,\mathrm{d}x\in[\delta_{q+1},400\delta_{q+1}].

(4.14)

The bounds for $\eta_{i}^{k}$ follow from (4.10). This value is chosen now so that the averaged rank-one tensor has the required normalization in the cancellation below.

Next, we define the trajectory of the center of the core as

\frac{\mathrm{d}}{\mathrm{d}t}x^{k}_{i}(t)=\frac{\eta_{i}^{k}\zeta_{i}^{k}(t)}{\mathfrak{R}_{i}^{k}(x^{k}_{i}(t))}\xi_{i},\qquad t\in\mathcal{T}_{i}^{k}.

(4.15)

For later use, set

R_{i}^{k}(t):=\mathfrak{R}_{i}^{k}(x_{i}^{k}(t)).

Set $t_{0}:=\tau_{q+1}(k+(i-1)/9+\lambda_{q+1}^{-1})$ and choose $x_{i}^{k}(t_{0})$ so that

\frac{1}{c_{i}\lambda^{2}_{q+1}}\int_{0}^{c_{i}\lambda_{q+1}^{2}}a_{i}^{k}(x_{i}^{k}(t_{0})+s\xi_{i})\,\mathrm{d}s=\int_{\mathbb{T}^{3}}a_{i}^{k}(x)\,\mathrm{d}x=\frac{2\pi(\eta_{i}^{k})^{2}}{9}.

Such a point exists: the orbit-average on the left is a continuous function on the connected quotient of $\mathbb{T}^{3}$ by the closed $\xi_{i}$ -orbit, and its average over that quotient is precisely the spatial mean of $a_{i}^{k}$ . It must therefore attain that mean.

Solve the ODE (4.15) forward and backward from $x_{i}^{k}(t_{0})$ to obtain a trajectory on the whole interval $\mathcal{T}_{i}^{k}$ . We impose

\displaystyle\lambda_{q+1}^{3}r_{q+1}\delta_{q+1}^{1/2}\leq\frac{\tau_{q+1}}{C_{\ast}},

(4.16)

where $C_{\ast}$ is a sufficiently large absolute constant. This condition is verified uniformly, including at $q=0$ , in Section 5.3.

On the plateau $\bar{\mathcal{T}}_{i}^{k}$ where $\zeta_{i}^{k}=1$ , the trajectory is periodic with period

T_{i}^{k}=\int_{0}^{c_{i}\lambda_{q+1}^{2}}\frac{r_{q+1}}{\eta_{i}^{k}}a_{i}^{k}(x_{i}^{k}(t_{0})+s\xi_{i})\,\mathrm{d}s=\frac{2\pi c_{i}\lambda_{q+1}^{2}r_{q+1}\eta_{i}^{k}}{9}\leq Cc_{i}\lambda_{q+1}^{2}r_{q+1}\delta_{q+1}^{1/2}\leq\frac{\tau_{q+1}}{10\lambda_{q+1}}.

(4.17)

Condition (4.16) is intended to guarantee that many complete periods lie inside $\bar{\mathcal{T}}_{i}^{k}$ . Let $M_{i}^{k}$ be the largest nonnegative integer such that

t_{0}+M_{i}^{k}T_{i}^{k}\leq\tau_{q+1}\left(k+\frac{i}{9}-\lambda_{q+1}^{-1}\right).

By the maximality of $M_{i}^{k}$ , the completed periods differ from $\tau_{q+1}/9$ by at most the two cutoff layers and one additional period:

0\leq\frac{1}{9}\tau_{q+1}-M_{i}^{k}T_{i}^{k}\leq\frac{2\tau_{q+1}}{\lambda_{q+1}}+T_{i}^{k}\leq C\frac{\tau_{q+1}}{\lambda_{q+1}}.

(4.18)

4.6 Estimates of the building block

We apply Corollary 3.2 with its intrinsic exponent set equal to $p_{0}$ from (4.1). We take profile direction $e=-\xi_{i}$ , amplitude $\eta_{i}^{k}\zeta_{i}^{k}$ , radius $\mathfrak{R}_{i}^{k}(x_{i}^{k}(t))$ , and center $x_{i}^{k}(t)$ . The sign is determined by the normalization of the Hill impulse: with this choice the resulting block $V_{i}^{k}$ moves and has impulse in the $+\xi_{i}$ direction. It satisfies

\partial_{t}V^{k}_{i}+\operatorname{div}(V^{k}_{i}\otimes V^{k}_{i})-\nu\Delta V^{k}_{i}+\nabla P^{k}_{i}=\frac{\mathrm{d}}{\mathrm{d}t}(\eta^{k}_{i}\zeta^{k}_{i}(t)R^{k}_{i}(t))\frac{1}{R^{k}_{i}(t)}\tilde{H}_{R^{k}_{i}(t),-\xi_{i}}(x-x_{i}^{k}(t))+\operatorname{div}(F^{k}_{i}).

(4.19)

With these choices of radius and amplitude, Proposition 3.1 and Corollary 3.2 give the following estimates. Since $\mathfrak{R}_{i}^{k}=r_{q+1}a_{i}^{k}$ ,

	$\displaystyle\\|\nabla\log\mathfrak{R}_{i}^{k}\\|_{L_{x}^{\infty}}$	$\displaystyle=\left\\|\frac{\nabla a_{i}^{k}}{a_{i}^{k}}\right\\|_{L_{x}^{\infty}}$
		$\displaystyle\leq C\delta_{q+1}^{-1}\lambda_{q}^{4n+4\gamma\sigma}\leq C\lambda_{q+1}^{\frac{4n}{\sigma}+5\gamma}.$

Using the bounds for $\eta_{i}^{k}$ , $R_{i}^{k}$ , and $\nabla\log\mathfrak{R}_{i}^{k}$ , we obtain

$\displaystyle\\|F_{i}^{k}\\|_{L_{t}^{\infty}L_{x}^{1}}$	$\displaystyle\leq C(\eta_{i}^{k})^{2}\\|R_{i}^{k}\\|_{L_{t}^{\infty}}^{\kappa}\left(1+\\|\nabla\log\mathfrak{R}_{i}^{k}\\|_{L_{x}^{\infty}}\right)$
	$\displaystyle\quad+C\nu\eta_{i}^{k}\\|R_{i}^{k}\\|_{L_{t}^{\infty}}^{\frac{2}{p_{0}}-\frac{5}{3}}$
	$\displaystyle\leq C\delta_{q+1}r_{q+1}^{\kappa}\lambda_{q+1}^{\kappa\left(\frac{3n}{\sigma}+3\gamma\right)+\frac{4n}{\sigma}+5\gamma}$
	$\displaystyle\quad+C\nu\delta_{q+1}^{1/2}r_{q+1}^{\frac{2}{p_{0}}-\frac{5}{3}}\lambda_{q+1}^{\left(\frac{3n}{\sigma}+3\gamma\right)\left(\frac{2}{p_{0}}-\frac{5}{3}\right)}.$	(4.20)

The velocity satisfies

	$\displaystyle\\|V_{i}^{k}\\|_{L_{t}^{\infty}L_{x}^{3/2}}$	$\displaystyle\leq C\eta_{i}^{k}\\|R_{i}^{k}\\|_{L_{t}^{\infty}}^{1/3}\leq C\delta_{q+1}^{1/2}r_{q+1}^{1/3}\lambda_{q+1}^{\frac{n}{\sigma}+\gamma},$		(4.21)
	$\displaystyle\\|V_{i}^{k}\\|_{L_{t}^{\infty}L_{x}^{2}}$	$\displaystyle\leq C\eta_{i}^{k}\leq C\delta_{q+1}^{1/2}.$		(4.22)

For $1<s\leq 6/5$ , the exponent $2/s-5/3$ is nonnegative. Using the upper bound for $R_{i}^{k}$ , we obtain

	$\displaystyle\\|\nabla V_{i}^{k}\\|_{L_{t}^{\infty}L_{x}^{s}}$	$\displaystyle\leq C_{s}\eta_{i}^{k}\left\\|(R_{i}^{k})^{\frac{2}{s}-\frac{5}{3}}\right\\|_{L_{t}^{\infty}}$
		$\displaystyle\leq C_{s}\delta_{q+1}^{1/2}r_{q+1}^{\frac{2}{s}-\frac{5}{3}}\lambda_{q+1}^{\left(\frac{3n}{\sigma}+3\gamma\right)\left(\frac{2}{s}-\frac{5}{3}\right)},\qquad 1<s\leq\frac{6}{5}.$		(4.23)

For $6/5<s<\infty$ , the exponent is negative. Using $R_{i}^{k}\geq r_{q+1}\delta_{q+1}$ , we instead obtain

	$\displaystyle\\|\nabla V_{i}^{k}\\|_{L_{t}^{\infty}L_{x}^{s}}$	$\displaystyle\leq C_{s}\eta_{i}^{k}\left\\|(R_{i}^{k})^{\frac{2}{s}-\frac{5}{3}}\right\\|_{L_{t}^{\infty}}$
		$\displaystyle\leq C_{s}\delta_{q+1}^{1/2}r_{q+1}^{\frac{2}{s}-\frac{5}{3}}\delta_{q+1}^{\frac{2}{s}-\frac{5}{3}},\qquad\frac{6}{5}<s<\infty.$		(4.24)

The following time-derivative bounds hold for every $s\in(1,\infty)$ .

$\displaystyle\\|\partial_{t}V^{k}_{i}\\|_{L^{\infty}_{t}L^{s}_{x}}$	$\displaystyle\leq C_{s}(\eta^{k}_{i})^{2}\\|(R^{k}_{i})^{-3}\\|_{L^{\infty}_{t}}\left(1+\\|\nabla\mathfrak{R}^{k}_{i}\\|_{L^{\infty}_{x}}\right)+C_{s}\eta^{k}_{i}\\|(\zeta^{k}_{i})^{\prime}\\|_{L^{\infty}_{t}}\\|(R^{k}_{i})^{-1}\\|_{L^{\infty}_{t}}$
	$\displaystyle\leq C\delta_{q+1}^{-2}r_{q+1}^{-3}(1+r_{q+1}\lambda_{q+1}^{4\frac{n}{\sigma}+4\gamma})+C\delta_{q+1}^{-\frac{1}{2}}(\lambda_{q+1}\tau_{q+1}^{-1})r_{q+1}^{-1}$
	$\displaystyle\leq C\delta_{q+1}^{-2}r_{q+1}^{-3}(1+r_{q+1}\lambda_{q+1}^{4\frac{n}{\sigma}+4\gamma}).$	(4.25)

The last inequality follows from (4.16). Similarly,

$\displaystyle\\|\partial_{t}\nabla V^{k}_{i}\\|_{L^{\infty}_{t}L^{s}_{x}}$	$\displaystyle\leq C_{s}(\eta^{k}_{i})^{2}\\|(R^{k}_{i})^{-3-\beta}\\|_{L^{\infty}_{t}}\left(1+\\|\nabla\mathfrak{R}^{k}_{i}\\|_{L^{\infty}_{x}}\right)+C_{s}\eta^{k}_{i}\\|(\zeta^{k}_{i})^{\prime}\\|_{L^{\infty}_{t}}\\|(R^{k}_{i})^{-\frac{5}{3}}\\|_{L^{\infty}_{t}}$
	$\displaystyle\leq C\delta_{q+1}^{-2-\beta}r_{q+1}^{-3-\beta}(1+r_{q+1}\lambda_{q+1}^{4\frac{n}{\sigma}+4\gamma})+C\delta_{q+1}^{-\frac{7}{6}}(\lambda_{q+1}\tau_{q+1}^{-1})r_{q+1}^{-\frac{5}{3}}$
	$\displaystyle\leq C\delta_{q+1}^{-2-\beta}r_{q+1}^{-3-\beta}(1+r_{q+1}\lambda_{q+1}^{4\frac{n}{\sigma}+4\gamma}).$	(4.26)

The last estimate again uses (4.16).

4.7 Auxiliary building block

The source generated by the moving Hill block has the correct spatial mean but remains concentrated at the core scale. To compare its time average with the prescribed rank-one stress, we replace it by an orbit-adapted auxiliary source. Its profile is normalized both in space and along the rational trajectory, so that its time average recovers $\operatorname{div}(a_{i}^{k}\xi_{i}\otimes\xi_{i})$ up to a small error tensor. The difference between the original and auxiliary sources will be handled by a symmetric antidivergence. For each $i,k$ , define

U_{i}^{k}(x,t):=\frac{\mathrm{d}}{\mathrm{d}t}\left(\eta_{i}^{k}\zeta_{i}^{k}(t)R_{i}^{k}(t)\right)\tilde{U}_{i}^{k}(x-x_{i}^{k}(t))\xi_{i},\quad t\in\mathcal{T}_{i}^{k}.

(4.27)

Extend $U_{i}^{k}$ by zero and set the locally finite sum

U_{q+1}:=\sum_{k\in\mathcal{K}_{q+1}}\sum_{i=1}^{9}U_{i}^{k}.

Each component $U_{i}^{k}$ replaces the corresponding concentrated Hill source

\frac{\mathrm{d}}{\mathrm{d}t}\left(\eta_{i}^{k}\zeta_{i}^{k}(t)R_{i}^{k}(t)\right)\frac{1}{R_{i}^{k}(t)}\tilde{H}_{R_{i}^{k}(t),-\xi_{i}}.

The auxiliary profile is chosen so that the antidivergence of the difference between these two sources is small. In addition, $\widetilde{U}_{i}^{k}$ has the following properties:

(i)

The profile has mass $2\pi$ , matching the impulse of $R^{-1}\widetilde{H}_{R,-\xi_{i}}$ :

$\int_{\mathbb{T}^{3}}\tilde{U}_{i}^{k}(x)\,\mathrm{d}x=2\pi\,,$ (4.28)

(ii)

For every $x\in\mathbb{T}^{3}$ , its full-orbit average satisfies

\frac{1}{c_{i}\lambda^{2}_{q+1}}\int_{0}^{c_{i}\lambda_{q+1}^{2}}\tilde{U}_{i}^{k}(x-(x_{i}^{k}(t_{0})+s\xi_{i}))\,\mathrm{d}s=2\pi\,,

(4.29)

(iii)

$\operatorname{supp}\widetilde{U}_{i}^{k}\subset B_{C\lambda_{q+1}^{-1}}(0)$ , and for every $p\in[1,\infty]$ ,

\|\tilde{U}_{i}^{k}\|_{L^{p}}\leq C(p)\lambda_{q+1}^{3-\frac{3}{p}}\,,\qquad\text{and}\qquad\|\nabla\tilde{U}_{i}^{k}\|_{L^{p}}\leq C(p)\lambda_{q+1}^{4-\frac{3}{p}}\,.

(4.30)

Only these normalization, orbit-average, and localization properties are used below. For completeness, we recall the standard normalized periodization construction from [14]. The special rational geodesics chosen in Lemma 4.2 are $C_{0}\lambda_{q+1}^{-1}$ -dense, with a uniform constant $C_{0}$ . Choose a nonnegative $\varphi\in C_{c}^{\infty}(B_{2C_{0}})$ that is strictly positive on $\overline{B}_{C_{0}}$ . We regard the rescaled bump below as a function on $\mathbb{T}^{3}$ by $\mathbb{Z}^{3}$ -periodization and set

\varphi_{\lambda_{q+1}}(x):=\lambda_{q+1}^{3}\varphi(\lambda_{q+1}x).

Define the orbit average without inserting the initial translation,

\Phi_{\lambda_{q+1}}(x):=\frac{1}{c_{i}\lambda_{q+1}^{2}}\int_{0}^{c_{i}\lambda_{q+1}^{2}}\varphi_{\lambda_{q+1}}(x-s\xi_{i})\,\mathrm{d}s,\qquad\widetilde{U}_{i}^{k}(x):=\frac{2\pi\varphi_{\lambda_{q+1}}(x)}{\Phi_{\lambda_{q+1}}(x)}.

(4.31)

The covering property of the orbit gives $C^{-1}\leq\Phi_{\lambda_{q+1}}\leq C$ and $\|\nabla\Phi_{\lambda_{q+1}}\|_{L^{\infty}}\leq C\lambda_{q+1}$ . Moreover, $\Phi_{\lambda_{q+1}}$ is invariant under translation along the $\xi_{i}$ -orbit. Consequently, averaging $\widetilde{U}_{i}^{k}(x-(x_{i}^{k}(t_{0})+s\xi_{i}))$ cancels the denominator in (4.31) and proves (4.29). Fubini’s theorem then yields (4.28), while the quotient rule and the support of $\varphi_{\lambda_{q+1}}$ give (4.30).

Proposition 4.3.

Let $U_{q+1}$ be the locally finite sum defined above. Then

\|U_{q+1}\|_{L^{\infty}_{t}L^{p}_{x}}+\lambda_{q+1}^{-1}\|\nabla_{x}U_{q+1}\|_{L^{\infty}_{t}L^{p}_{x}}\leq C(p)\lambda_{q+1}^{3-\frac{3}{p}}\sup_{i,k}\|a^{k}_{i}\|_{C^{1}}.

(4.32)

Moreover, for every $i$ and $k$ , there exists a smooth symmetric tensor $G_{i}^{k}=G_{i}^{k}(x)$ such that

\frac{1}{\tau_{q+1}}\int_{\mathcal{T}^{k}}U_{q+1}(x,t)\,\mathrm{d}t=\sum_{i=1}^{9}\operatorname{div}\left(a_{i}^{k}(x)\xi_{i}\otimes\xi_{i}+G_{i}^{k}(x)\right),

(4.33)

\displaystyle\|G_{i}^{k}\|_{L_{x}^{1}}\leq C\frac{\|a_{i}^{k}\|_{C^{1}}}{\lambda_{q+1}}\,.

(4.34)

Proof of Proposition 4.3.

The definition (4.27) of $U_{i}^{k}$ and (4.30) imply, for every $t\in\mathcal{T}_{i}^{k}$ ,

	$\displaystyle\\|U_{i}^{k}(t)\\|_{L^{p}_{x}}+\lambda_{q+1}^{-1}\\|\nabla_{x}U_{i}^{k}(t)\\|_{L^{p}_{x}}$	$\displaystyle\leq\sup_{s\in\mathcal{T}^{k}_{i}}\Big\|\frac{\mathrm{d}}{\mathrm{d}s}(\eta_{i}^{k}\zeta_{i}^{k}(s)R_{i}^{k}(s))\Big\|\Big(\\|\tilde{U}_{i}^{k}\\|_{L^{p}_{x}}+\lambda_{q+1}^{-1}\\|\nabla\tilde{U}_{i}^{k}\\|_{L^{p}_{x}}\Big)$
		$\displaystyle\leq C(p)\lambda_{q+1}^{3-\frac{3}{p}}\sup_{s\in\mathcal{T}^{k}_{i}}\Big\|\frac{\mathrm{d}}{\mathrm{d}s}(\eta_{i}^{k}\zeta_{i}^{k}(s)R_{i}^{k}(s))\Big\|.$		(4.35)

Using (4.12), (4.14), and (4.11), we estimate the supremum in (4.7), for every $s\in\mathcal{T}_{i}^{k}$ , by

$\displaystyle\Big\|\frac{\mathrm{d}}{\mathrm{d}s}(\eta_{i}^{k}\zeta_{i}^{k}(s)R_{i}^{k}(s))\Big\|$	$\displaystyle\leq\Big\|\eta_{i}^{k}R_{i}^{k}(s)\frac{\mathrm{d}\zeta_{i}^{k}}{\mathrm{d}s}\Big\|+\Big\|(\eta^{k}_{i}\zeta^{k}_{i})^{2}\frac{\nabla R^{k}_{i}}{R^{k}_{i}}\Big\|$
	$\displaystyle\leq C\delta_{q+1}^{\frac{1}{2}}r_{q+1}\\|a^{k}_{i}\\|_{L^{\infty}_{x}}\,\frac{\lambda_{q+1}}{\tau_{q+1}}+C\delta_{q+1}\\|\frac{\nabla a_{i}^{k}}{a_{i}^{k}}\\|_{L^{\infty}_{x}}$
	$\displaystyle\leq C\\|a^{k}_{i}\\|_{C^{1}_{x}}\,.$	(4.36)

For the second term we used $a_{i}^{k}\geq\delta_{q+1}$ , while (4.16) controls the first term. Combining this estimate with (4.7) proves (4.32).

To prove (4.33), it suffices to establish

\displaystyle\frac{1}{\tau_{q+1}}\int_{\mathcal{T}_{i}^{k}}U_{i}^{k}(x,t)\,\mathrm{d}t=\operatorname{div}\left(a_{i}^{k}(x)\xi_{i}\otimes\xi_{i}+G_{i}^{k}(x)\right)\,,\quad x\in\mathbb{T}^{3}\,,

and then sum over $i=1,\dots,9$ . Integration by parts gives

$\displaystyle\int_{\mathcal{T}_{i}^{k}}U_{i}^{k}(x,t)\,\mathrm{d}t$	$\displaystyle=\int_{\mathcal{T}_{i}^{k}}\frac{\mathrm{d}}{\mathrm{d}t}\left(\eta_{i}^{k}\zeta_{i}^{k}(t)R_{i}^{k}(t)\right)\tilde{U}_{i}^{k}(x-x_{i}^{k}(t))\xi_{i}\,\mathrm{d}t$
	$\displaystyle=-\int_{\mathcal{T}_{i}^{k}}\eta_{i}^{k}\zeta_{i}^{k}(t)R_{i}^{k}(t)\frac{\mathrm{d}}{\mathrm{d}t}\left(\tilde{U}_{i}^{k}(x-x_{i}^{k}(t))\right)\,\xi_{i}\,\mathrm{d}t$
	$\displaystyle=\int_{\mathcal{T}_{i}^{k}}\left(\eta_{i}^{k}\zeta_{i}^{k}(t)\right)^{2}\operatorname{div}\left(\tilde{U}_{i}^{k}(x-x_{i}^{k}(t))\xi_{i}\otimes\xi_{i}\right)\,\mathrm{d}t$
	$\displaystyle=\operatorname{div}\left(\xi_{i}\otimes\xi_{i}\int_{\mathcal{T}_{i}^{k}}\left(\eta_{i}^{k}\zeta_{i}^{k}(t)\right)^{2}\tilde{U}_{i}^{k}(x-x_{i}^{k}(t))\,\mathrm{d}t\right)\,.$	(4.37)

Recall $t_{0}$ , $T_{i}^{k}$ , and $M_{i}^{k}$ from Section 4.5, and set $\widetilde{\mathcal{T}}_{i}^{k}=[t_{0},t_{0}+M_{i}^{k}T_{i}^{k}]$ . By (4.9), $\widetilde{\mathcal{T}}_{i}^{k}\subseteq\bar{\mathcal{T}}_{i}^{k}$ , and hence $\zeta_{i}^{k}(t)=1$ for $t\in\widetilde{\mathcal{T}}_{i}^{k}$ .

Next, we write

	$\displaystyle\int_{\mathcal{T}_{i}^{k}}$	$\displaystyle\left(\eta_{i}^{k}\zeta_{i}^{k}(t)\right)^{2}\tilde{U}_{i}^{k}(x-x_{i}^{k}(t))\,\mathrm{d}t$
		$\displaystyle=\big(\eta_{i}^{k}\big)^{2}\int_{\tilde{\mathcal{T}}_{i}^{k}}\tilde{U}_{i}^{k}(x-x_{i}^{k}(t))\,\mathrm{d}t+\big(\eta_{i}^{k}\big)^{2}\int_{\mathcal{T}_{i}^{k}\setminus\tilde{\mathcal{T}}_{i}^{k}}\zeta_{i}^{k}(t)^{2}\tilde{U}_{i}^{k}(x-x_{i}^{k}(t))\,\mathrm{d}t\eqqcolon I+II\,.$		(4.38)

The tensor $II\,\xi_{i}\otimes\xi_{i}$ contributes to $G_{i}^{k}$ . By (4.30),

\|II\|_{L^{1}}\leq(\eta_{i}^{k})^{2}\|\tilde{U}_{i}^{k}\|_{L^{1}}\int_{\mathcal{T}_{i}^{k}\setminus\tilde{\mathcal{T}}_{i}^{k}}(\zeta_{i}^{k}(t))^{2}\,\mathrm{d}t\leq C\delta_{q+1}\tau_{q+1}\lambda_{q+1}^{-1}\,.

(4.39)

Next, we investigate the main term $I$ . Make the change of variables $t=t(s)\in\widetilde{\mathcal{T}}_{i}^{k}$ , determined by $x_{i}^{k}(t(s))=x_{i}^{k}(t_{0})+s\xi_{i}$ and $t(0)=t_{0}$ . Then

t^{\prime}(s)=\frac{r_{q+1}}{\eta_{i}^{k}\zeta_{i}^{k}(t(s))}a_{i}^{k}(x_{i}^{k}(t_{0})+s\xi_{i})\,.

This change of variables gives

$\displaystyle I$	$\displaystyle=\eta_{i}^{k}r_{q+1}\int_{0}^{c_{i}M_{i}^{k}\lambda_{q+1}^{2}}a_{i}^{k}(x_{i}^{k}(t_{0})+s\xi_{i})\tilde{U}_{i}^{k}(x-(x_{i}^{k}(t_{0})+s\xi_{i}))\,\mathrm{d}s$
	$\displaystyle=\eta_{i}^{k}r_{q+1}a_{i}^{k}(x)\int_{0}^{c_{i}M_{i}^{k}\lambda_{q+1}^{2}}\tilde{U}_{i}^{k}(x-(x_{i}^{k}(t_{0})+s\xi_{i}))\,\mathrm{d}s$
	$\displaystyle+\eta_{i}^{k}r_{q+1}\int_{0}^{c_{i}M_{i}^{k}\lambda_{q+1}^{2}}(a_{i}^{k}(x_{i}^{k}(t_{0})+s\xi_{i})-a_{i}^{k}(x))\tilde{U}_{i}^{k}(x-(x_{i}^{k}(t_{0})+s\xi_{i}))\,\mathrm{d}s$
	$\displaystyle\eqqcolon I^{\prime}+II^{\prime},$	(4.40)

where $I^{\prime}$ is the main term and $II^{\prime}$ will be part of the error $G^{k}_{i}$ . By (4.29), we rewrite the main term as

	$\displaystyle I^{\prime}$	$\displaystyle=\eta_{i}^{k}r_{q+1}a_{i}^{k}(x)\int_{0}^{c_{i}M_{i}^{k}\lambda_{q+1}^{2}}\widetilde{U}_{i}^{k}(x-(x_{i}^{k}(t_{0})+s\xi_{i}))\,\mathrm{d}s$
		$\displaystyle=\eta_{i}^{k}r_{q+1}2\pi c_{i}M_{i}^{k}\lambda_{q+1}^{2}a_{i}^{k}(x)=(\tau_{q+1}+E_{i}^{k})a_{i}^{k}(x),$		(4.41)

where $E_{i}^{k}:=9T_{i}^{k}M_{i}^{k}-\tau_{q+1}$ . By (4.17) and (4.18),

|E_{i}^{k}|\leq C\frac{\tau_{q+1}}{\lambda_{q+1}}.

(4.42)

The term $II^{\prime}$ in (4.40) satisfies

\displaystyle\|II^{\prime}\|_{L^{1}}\leq C\eta_{i}^{k}r_{q+1}2\pi c_{i}M_{i}^{k}\lambda^{2}_{q+1}\frac{\|a_{i}^{k}\|_{C^{1}}}{\lambda_{q+1}}\leq C\left(\tau_{q+1}+|E_{i}^{k}|\right)\frac{\|a_{i}^{k}\|_{C^{1}}}{\lambda_{q+1}}\leq C\tau_{q+1}\frac{\|a_{i}^{k}\|_{C^{1}}}{\lambda_{q+1}}.

(4.43)

Combining (4.37), (4.38), (4.40), and (4.41), we obtain

	$\displaystyle\frac{1}{\tau_{q+1}}\int_{\mathcal{T}_{i}^{k}}U_{i}^{k}(x,t)\,\mathrm{d}t$	$\displaystyle=\operatorname{div}\left(a_{i}^{k}(x)\xi_{i}\otimes\xi_{i}+G_{i}^{k}(x)\right),$
	$\displaystyle G_{i}^{k}(x)$	$\displaystyle:=\frac{1}{\tau_{q+1}}\left(II+II^{\prime}+E_{i}^{k}a_{i}^{k}(x)\right)\xi_{i}\otimes\xi_{i}.$

Finally, (4.39), (4.42), and (4.43) give

\displaystyle\|G^{k}_{i}\|_{L^{1}}\leq C\frac{\delta_{q+1}}{\lambda_{q+1}}+C\frac{\|a_{i}^{k}\|_{C^{1}}}{\lambda_{q+1}}+C\frac{\|a_{i}^{k}\|_{L^{1}}}{\lambda_{q+1}}\leq C\frac{\|a_{i}^{k}\|_{C^{1}}}{\lambda_{q+1}}.

∎

4.8 The temporal corrector

The auxiliary source cancels the Reynolds stress only after averaging over a complete time cell. We therefore introduce a temporal corrector as the primitive of its zero-mean oscillatory part. The cellwise cancellation confines this primitive to an interval of length at most $\tau_{q+1}$ , producing the small factor needed in the estimates, while the Leray projection preserves incompressibility. Here $\mathbb{P}=\operatorname{Id}-\nabla\Delta^{-1}\operatorname{div}$ , where $\Delta^{-1}$ is the inverse Laplacian on the periodic domain $\mathbb{T}^{3}$ , acting on mean-zero scalar functions. Define $Q_{q+1}:\mathbb{T}^{3}\times I_{\ast}\to\mathbb{R}^{3}$ by

\displaystyle-Q_{q+1}(x,t)\coloneqq\mathbb{P}\int_{k\tau_{q+1}}^{t}\left(U_{q+1}(x,s)-\frac{1}{\tau_{q+1}}\int_{\mathcal{T}^{k}}U_{q+1}(x,z)\,\mathrm{d}z\right)\mathrm{d}s,\qquad t\in\mathcal{T}^{k}.

(4.44)

The integrand in (4.44) has zero integral over every complete cell $[k\tau_{q+1},(k+1)\tau_{q+1}]$ . Consequently, only the current cell contributes to the primitive, and its effective time length is at most $\tau_{q+1}$ .

-Q_{q+1}(x,t)=\mathbb{P}\int_{k\tau_{q+1}}^{t}\left(U_{q+1}(x,s)-\frac{1}{\tau_{q+1}}\int_{\mathcal{T}^{k}}U_{q+1}(x,\tau)\mathrm{d}\tau\right)\,\mathrm{d}s.

(4.45)

In particular, $Q_{q+1}$ and $\nabla Q_{q+1}$ vanish at every cell interface. Their time derivatives may have bounded jumps there, so $Q_{q+1}$ is continuous and piecewise smooth, in agreement with the admissibility convention at the beginning of this section. Since the velocity itself is continuous, these jumps create no temporal Dirac masses in the distributional equation.

The norms of $Q_{q+1}$ are estimated using (4.32). By (4.45) and the Calderón–Zygmund bounds for $\mathbb{P}$ ,

	$\displaystyle\\|Q_{q+1}\\|_{L_{t}^{\infty}L_{x}^{p}}$	$\displaystyle\leq C\int_{k\tau_{q+1}}^{t}\\|U_{q+1}(\cdot,s)\\|_{L_{x}^{p}}\,\mathrm{d}s\leq C\tau_{q+1}\\|U_{q+1}\\|_{L_{t}^{\infty}L_{x}^{p}},$
	$\displaystyle\\|\nabla Q_{q+1}\\|_{L_{t}^{\infty}L_{x}^{p}}$	$\displaystyle\leq C\int_{k\tau_{q+1}}^{t}\\|\nabla U_{q+1}(\cdot,s)\\|_{L_{x}^{p}}\,\mathrm{d}s\leq C\tau_{q+1}\\|\nabla_{x}U_{q+1}\\|_{L_{t}^{\infty}L_{x}^{p}}.$		(4.46)

These estimates hold for every $p\in(1,\infty)$ and $t\in\mathcal{T}^{k}$ . Equation (4.44) also gives

	$\displaystyle\partial_{t}Q_{q+1}(x,t)$	$\displaystyle=-\mathbb{P}U_{q+1}(x,t)+\frac{1}{\tau_{q+1}}\int_{\mathcal{T}^{k}}\mathbb{P}U_{q+1}(x,s)\,\mathrm{d}s,$
	$\displaystyle\\|\partial_{t}Q_{q+1}\\|_{L_{t}^{\infty}L_{x}^{p}}$	$\displaystyle\leq 2\\|\mathbb{P}U_{q+1}\\|_{L_{t}^{\infty}L_{x}^{p}}\leq C\\|U_{q+1}\\|_{L_{t}^{\infty}L_{x}^{p}}.$		(4.47)

Similarly,

\displaystyle\|\partial_{t}\nabla Q_{q+1}\|_{L^{\infty}_{t}L^{p}_{x}}\leq 2\|\nabla\mathbb{P}U_{q+1}\|_{L^{\infty}_{t}L^{p}_{x}}\leq C\|\nabla U_{q+1}\|_{L^{\infty}_{t}L^{p}_{x}}.

(4.48)

4.9 Construction of the next Reynolds flow

All components of the perturbation are now available. We add the moving Hill blocks and the temporal corrector to the mollified background flow. Substitution into the equation produces the next Navier–Stokes–Reynolds system, with the remaining terms grouped into the linear interaction, corrector, temporal-freezing, source-replacement, and building-block errors.

Define the velocity field $u_{q+1}$ by

u_{q+1}(x,t)\coloneqq u_{l}(x,t)+v_{q+1}(x,t)+Q_{q+1}(x,t)\,,\quad\quad x\in\mathbb{T}^{3}\,,\,\,t\in I_{\ast}\,.

(4.49)

Here the main velocity perturbation is

v_{q+1}(x,t)=\sum_{k\in\mathcal{K}_{q+1}}\sum_{i=1}^{9}V_{i}^{k}(x,t).

The fields $V_{i}^{k}$ are the adapted building blocks, and $Q_{q+1}$ is the temporal corrector constructed above.

Remark 4.4.

For every grid index $k\in\{0,\dots,\tau_{q+1}^{-1}\}$ , $v_{q+1}(x,k\tau_{q+1})=Q_{q+1}(x,k\tau_{q+1})=0$ by the cutoff and (4.45). Thus

u_{q+1}(x,k\tau_{q+1})=u_{l}(x,k\tau_{q+1}),\quad\text{for every grid index $k$ and $x\in\mathbb{T}^{3}$}\,.

In particular, for $j\in\{0,1\}$ ,

\|u_{q+1}(\cdot,j)-u_{q}(\cdot,j)\|_{L^{2}}\leq\|u_{q}-u_{l}\|_{L_{t}^{\infty}L_{x}^{2}}\leq Cl\|(\nabla_{x}u_{q},\partial_{t}u_{q})\|_{L_{t}^{\infty}L_{x}^{2}}\leq Cc_{0}\lambda_{q+1}^{-\gamma}=Cc_{0}\lambda_{q}^{-1/5}.

(4.50)

We define the new pressure field as

	$\displaystyle\nabla p_{q+1}\coloneqq{}$	$\displaystyle\nabla p_{l}-\nabla\zeta+\sum_{k\in\mathcal{K}_{q+1}}\sum_{i=1}^{9}\nabla P_{i}^{k}$
		$\displaystyle-(\mathrm{Id}-\mathbb{P})\left(U_{q+1}(x,t)-\frac{1}{\tau_{q+1}}\int_{\mathcal{T}^{k}}U_{q+1}(x,\tau)\,\mathrm{d}\tau\right),\qquad t\in\mathcal{T}^{k}.$

Here $p_{l}$ , $\zeta$ , and $P_{i}^{k}$ are defined in (4.4), (4.5), and (4.19), respectively. The last term is the gradient part of the temporal-corrector source. The locally finite pressure sum is defined up to a function of time. The fields $u_{q+1}$ and $p_{q+1}$ satisfy the Navier–Stokes–Reynolds system distributionally with stress

\mathcal{E}_{q+1}\coloneqq\mathcal{E}_{q+1}^{(l)}+\mathcal{E}_{q+1}^{(c)}+\mathcal{E}_{q+1}^{(t)}+\mathcal{E}_{q+1}^{(s)}+\sum_{i=1}^{9}(F_{i}^{k}+G_{i}^{k}),\qquad t\in\mathcal{T}^{k}.

(4.51)

Here $F_{i}^{k}$ is defined in (4.19) and $G_{i}^{k}$ in Proposition 4.3. The stress $\mathcal{E}_{q+1}^{(l)}$ contains terms linear in the principal perturbation, whereas $\mathcal{E}_{q+1}^{(c)}$ contains the temporal-corrector terms:

	$\displaystyle\mathcal{E}_{q+1}^{(l)}$	$\displaystyle\coloneqq v_{q+1}\otimes u_{l}+u_{l}\otimes v_{q+1}\,,$
	$\displaystyle\mathcal{E}_{q+1}^{(c)}$	$\displaystyle\coloneqq Q_{q+1}\otimes(u_{l}+v_{q+1})+(u_{l}+v_{q+1})\otimes Q_{q+1}+Q_{q+1}\otimes Q_{q+1}-\nu(\nabla Q_{q+1}+(\nabla Q_{q+1})^{T})\,.$

Finally, for $t\in\mathcal{T}^{k}$ , define $\mathcal{E}_{q+1}^{(t)}$ as the coefficient-freezing error and $\mathcal{E}_{q+1}^{(s)}$ as the source-replacement error:

\displaystyle\mathcal{E}_{q+1}^{(t)}(x,t)

\displaystyle\coloneqq\sum_{i=1}^{9}(a_{i}^{k}(x)-a_{i}(x,t))\xi_{i}\otimes\xi_{i},\qquad t\in\mathcal{T}^{k},

	$\displaystyle S_{i}^{k}(x,t)$	$\displaystyle\coloneqq\left[\frac{1}{R_{i}^{k}(t)}\widetilde{H}_{R_{i}^{k}(t),-\xi_{i}}(\,x-x_{i}^{k}(t))-\widetilde{U}_{i}^{k}(\,x-x_{i}^{k}(t))\xi_{i}\right]$
		$\displaystyle\quad\times\frac{\mathrm{d}}{\mathrm{d}t}\bigl(\eta_{i}^{k}\zeta_{i}^{k}(t)R_{i}^{k}(t)\bigr),$
	$\displaystyle\mathcal{E}_{q+1}^{(s)}(x,t)$	$\displaystyle\coloneqq\sum_{i=1}^{9}\mathcal{R}(S_{i}^{k}(\cdot,t))(x),\qquad t\in\mathcal{T}^{k}.$

The impulse identity for the Hill source and (4.28) give

\int_{\mathbb{T}^{3}}S_{i}^{k}(x,t)\,\mathrm{d}x=0,

so the standard symmetric inverse-divergence $\mathcal{R}$ on the torus is applicable. By (4.13) and property (iii) of $\widetilde{U}_{i}^{k}$ , both terms are supported in a common ball of radius $C\lambda_{q+1}^{-1}$ . Poincaré’s inequality on that ball gives $\|S_{i}^{k}\|_{W^{-1,p}}\leq C_{p}\lambda_{q+1}^{-1}\|S_{i}^{k}\|_{L^{p}}$ ; the order- $-1$ estimate for $\mathcal{R}$ therefore yields, for $1<p<\infty$ ,

\operatorname{div}\mathcal{R}S_{i}^{k}=S_{i}^{k},\qquad\|\mathcal{R}S_{i}^{k}\|_{L^{p}}\leq C_{p}\lambda_{q+1}^{-1}\|S_{i}^{k}\|_{L^{p}}.

Finally, substituting (4.19), (4.44), (4.33), and (4.5) into (4.49) gives (4.2) with the stress (4.51); the signs of $\mathcal{E}_{q+1}^{(t)}$ and of the pressure correction $-\nabla\zeta$ are exactly those dictated by (4.5).

5 Proof of Proposition 4.1

We verify the conclusions of Proposition 4.1. We first control the velocity increment and its space–time regularity, then estimate each component of the new Reynolds stress, and finally choose the parameters so that all the resulting inequalities hold simultaneously.

5.1 Velocity estimates

We begin with the bounds for the new velocity. The $L^{2}$ estimate controls the size of the increment, the $C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}$ estimate propagates the target gradient regularity, and the derivative estimates preserve the inductive $L^{4}$ bound. Each estimate yields an explicit restriction on the iteration parameters. By (4.50), (4.22), and (4.8),

At each time, at most one pair $(i,k)$ is active. Hence the norm of the sum of moving blocks is controlled by the supremum over $(i,k)$ , as used below.

	$\displaystyle\\|u_{q+1}-u_{q}\\|_{L_{t}^{\infty}L_{x}^{2}}$	$\displaystyle\leq\\|u_{l}-u_{q}\\|_{L_{t}^{\infty}L_{x}^{2}}+\sup_{i,k}\\|V_{i}^{k}\\|_{L_{t}^{\infty}L_{x}^{2}}+\\|Q_{q+1}\\|_{L_{t}^{\infty}L_{x}^{2}}$
		$\displaystyle\leq C\lambda_{q+1}^{-\gamma}+C\delta_{q+1}^{\frac{1}{2}}+C\tau_{q+1}\\|U_{q+1}\\|_{L_{t}^{\infty}L_{x}^{2}}$
		$\displaystyle\leq C\lambda_{q+1}^{-\gamma}+C\lambda_{1}^{\gamma}\lambda_{q+1}^{-\frac{\gamma}{2}}+C\tau_{q+1}\lambda_{q+1}^{\frac{3}{2}+4\frac{n}{\sigma}+4\gamma}.$

To obtain $\|u_{q+1}-u_{q}\|_{L_{t}^{\infty}L_{x}^{2}}\leq C\delta_{q+1}^{1/2}=C\lambda_{1}^{\gamma}\lambda_{q+1}^{-\gamma/2}$ , it is sufficient that

\lambda_{q+1}^{-\eta+\frac{3}{2}+4\gamma+4\frac{n}{\sigma}}\leq\lambda_{q+1}^{-\frac{\gamma}{2}}\leq\delta_{q+1}^{1/2}.

(5.1)

Moreover,

\|u_{q+1}\|_{L_{t}^{\infty}L_{x}^{2}}\leq 2\delta_{0}^{\frac{1}{2}}-\delta_{q}^{\frac{1}{2}}+\|u_{q+1}-u_{q}\|_{L_{t}^{\infty}L_{x}^{2}}\leq 2\delta_{0}^{\frac{1}{2}}-\delta_{q}^{\frac{1}{2}}+C\delta_{q+1}^{\frac{1}{2}}\leq 2\delta_{0}^{\frac{1}{2}}-\delta_{q+1}^{\frac{1}{2}},

provided that $\lambda_{0}$ is sufficiently large.

Remark 5.1.

For $2<s<4$ , the same argument gives the following $L_{t}^{\infty}L_{x}^{s}$ -bound:

	$\displaystyle\\|u_{q+1}-u_{q}\\|_{L_{t}^{\infty}L_{x}^{s}}$	$\displaystyle\leq\\|u_{l}-u_{q}\\|_{L_{t}^{\infty}L_{x}^{s}}+\sup_{i,k}\\|V_{i}^{k}\\|_{L_{t}^{\infty}L_{x}^{s}}+\\|Q_{q+1}\\|_{L_{t}^{\infty}L_{x}^{s}}$
		$\displaystyle\leq Cl\\|(\nabla_{x}u_{q},\partial_{t}u_{q})\\|_{L_{t}^{\infty}L_{x}^{4}}+C\delta_{q+1}^{\frac{1}{2}}\\|R^{\frac{2}{s}-1}\\|_{L_{t}^{\infty}}+C\tau_{q+1}\\|U_{q+1}\\|_{L_{t}^{\infty}L_{x}^{s}}$
		$\displaystyle\leq C\lambda_{q+1}^{-\gamma}+C\lambda_{1}^{(\frac{4}{s}-1)\gamma}\lambda_{q+1}^{-\frac{\gamma}{2}+(1-\frac{2}{s})(\gamma+\mu)}+C\lambda_{q+1}^{-\eta+3-\frac{3}{s}+4\frac{n}{\sigma}+4\gamma}.$

Since $I_{\ast}$ has finite length, this also implies, for every time exponent $1\leq r\leq\infty$ ,

\|u_{q+1}-u_{q}\|_{L_{t}^{r}(I_{\ast};L_{x}^{s})}\leq|I_{\ast}|^{1/r}\|u_{q+1}-u_{q}\|_{L_{t}^{\infty}(I_{\ast};L_{x}^{s})}.

This formulation avoids incorrectly replacing the $L_{t}^{r}$ -norm of the sum over all time cells by the supremum of the individual cellwise $L_{t}^{r}$ -norms.

The perturbation also preserves the zero-mean condition. Indeed, Proposition 3.1 and the choice $e=-\xi_{i}$ give

\int_{\mathbb{T}^{3}}V_{i}^{k}(x,t)\,\mathrm{d}x=2\pi\eta_{i}^{k}\zeta_{i}^{k}(t)R_{i}^{k}(t)\xi_{i}.

On the other hand, (4.28) and (4.27) show that $\int_{\mathbb{T}^{3}}U_{i}^{k}\,\mathrm{d}x=2\pi\frac{\mathrm{d}}{\mathrm{d}t}(\eta_{i}^{k}\zeta_{i}^{k}R_{i}^{k})\xi_{i}$ . The cell average of this spatial mean vanishes because $\zeta_{i}^{k}$ vanishes at the endpoints of its support. Taking the spatial mean in (4.44) therefore yields

\int_{\mathbb{T}^{3}}Q_{q+1}(x,t)\,\mathrm{d}x=-2\pi\eta_{i}^{k}\zeta_{i}^{k}(t)R_{i}^{k}(t)\xi_{i}

on the active subinterval, and zero elsewhere. Thus $v_{q+1}+Q_{q+1}$ has zero spatial mean; since mollification preserves the mean, so does $u_{q+1}$ .

We next estimate $\nabla u_{q+1}$ . By (4.23) and (4.8),

	$\displaystyle\\|\nabla u_{q+1}\\|_{L_{t}^{\infty}L_{x}^{6/5}}$	$\displaystyle\leq\\|\nabla u_{l}\\|_{L_{t}^{\infty}L_{x}^{6/5}}+\sup_{i,k}\\|\nabla V_{i}^{k}\\|_{L_{t}^{\infty}L_{x}^{6/5}}+\\|\nabla Q_{q+1}\\|_{L_{t}^{\infty}L_{x}^{6/5}}$
		$\displaystyle\leq\\|\nabla u_{q}\\|_{L_{t}^{\infty}L_{x}^{6/5}}+C\delta_{q+1}^{\frac{1}{2}}+C\tau_{q+1}\lambda_{q+1}^{\frac{3}{2}+4\frac{n}{\sigma}+4\gamma}\,.$

The desired gain by $C\delta_{q+1}^{\epsilon}$ follows provided that

-\eta+\frac{3}{2}+4\frac{n}{\sigma}+4\gamma<0.

(5.2)

Remark 5.2.

For an exponent slightly larger than $6/5$ , the power of $R_{i}^{k}$ is negative; hence one must use (4.24), rather than (4.23). Because $r_{q+1}$ is much smaller than $\delta_{q+1}$ , this loss restricts the attainable exponent. More precisely,

	$\displaystyle\\|\nabla u_{q+1}\\|_{L_{t}^{\infty}L_{x}^{\bar{p}}}$	$\displaystyle\leq\\|\nabla u_{l}\\|_{L_{t}^{\infty}L_{x}^{\bar{p}}}+\sup_{i,k}\\|\nabla V_{i}^{k}\\|_{L_{t}^{\infty}L_{x}^{\bar{p}}}+\\|\nabla Q_{q+1}\\|_{L_{t}^{\infty}L_{x}^{\bar{p}}}$
		$\displaystyle\leq\\|\nabla u_{q}\\|_{L_{t}^{\infty}L_{x}^{\bar{p}}}+C\delta_{q+1}^{\frac{1}{2}}r_{q+1}^{\frac{2}{\bar{p}}-\frac{5}{3}}\delta_{q+1}^{\frac{2}{\bar{p}}-\frac{5}{3}}+C\tau_{q+1}\lambda_{q+1}^{4-\frac{3}{\bar{p}}+4\frac{n}{\sigma}+4\gamma}\,.$

A gain by $\delta_{q+1}^{\epsilon}$ follows if

-\frac{\gamma}{2}+(\frac{5}{3}-\frac{2}{\bar{p}})(\mu+\gamma)<0,\quad-\eta+4-\frac{3}{\bar{p}}+4\frac{n}{\sigma}+4\gamma<0.

(5.3)

Taking $s=4$ instead gives

\|\nabla u_{q+1}\|_{L_{t}^{\infty}L_{x}^{4}}\leq\|\nabla u_{q}\|_{L_{t}^{\infty}L_{x}^{4}}+C\delta_{q+1}^{\frac{1}{2}}r_{q+1}^{-\frac{7}{6}}\delta_{q+1}^{-\frac{7}{6}}+C\tau_{q+1}\lambda_{q+1}^{\frac{13}{4}+4\frac{n}{\sigma}+4\gamma}\,.

To keep the induction assumption $\|\nabla u_{q+1}\|_{L_{t}^{\infty}L_{x}^{4}}\leq\lambda_{q+1}^{n}$ , we need

-\frac{\gamma}{2}+\frac{7}{6}(\mu+\gamma)<n,\quad-\eta+\frac{13}{4}+4\frac{n}{\sigma}+4\gamma<n.

(5.4)

We now justify the Hölder continuity in time without hiding any exponent loss. Set

	$\displaystyle D_{q+1}$	$\displaystyle:=\delta_{q+1}^{-2-\beta}r_{q+1}^{-3-\beta}\left(1+r_{q+1}\lambda_{q+1}^{4n/\sigma+4\gamma}\right),$
	$\displaystyle B$	$\displaystyle:=\max\left\{(2+\beta)\gamma+(3+\beta)\mu,\,(2+\beta)\gamma+(2+\beta)\mu+\frac{4n}{\sigma}+4\gamma\right\}.$

Thus $D_{q+1}\leq C\lambda_{q+1}^{B}$ . For every Banach-valued function $f$ that is continuous and piecewise $C^{1}$ ,

[f]_{C_{t}^{\alpha_{0}}}\leq C\|f\|_{L_{t}^{\infty}}^{1-\alpha_{0}}\|\partial_{t}f\|_{L_{t}^{\infty}}^{\alpha_{0}}.

This estimate applies globally here: the supports of the $V_{i}^{k}$ are disjoint in time, each $\nabla V_{i}^{k}$ vanishes at the endpoints of its support, and $\nabla Q_{q+1}$ vanishes at the cell interfaces. Using (4.23), (4.26), (4.8), and (4.48), we obtain

	$\displaystyle[\nabla(v_{q+1}+Q_{q+1})]_{C_{t}^{\alpha_{0}}L_{x}^{6/5}}$	$\displaystyle\leq C\delta_{q+1}^{\frac{1}{2}(1-\alpha_{0})}D_{q+1}^{\alpha_{0}}+C\tau_{q+1}^{1-\alpha_{0}}\lambda_{q+1}^{\frac{3}{2}+4n/\sigma+4\gamma}$
		$\displaystyle\leq C\lambda_{1}^{\gamma(1-\alpha_{0})}\lambda_{q+1}^{-\frac{\gamma}{2}(1-\alpha_{0})+B\alpha_{0}}+C\lambda_{q+1}^{-\eta+\frac{3}{2}+4n/\sigma+4\gamma+\eta\alpha_{0}}.$

Both exponents are negative when $\alpha_{0}>0$ is sufficiently small, by (5.2).

For the target exponent $\bar{p}$ , set $\kappa_{\bar{p}}:=\frac{5}{3}-\frac{2}{\bar{p}}$ and introduce the two positive margins

\theta_{V}:=\frac{\gamma}{2}-\kappa_{\bar{p}}(\mu+\gamma),\qquad\theta_{Q}:=\eta-4+\frac{3}{\bar{p}}-\frac{4n}{\sigma}-4\gamma.

Their positivity is precisely (5.3). The same interpolation then gives

\displaystyle[\nabla(v_{q+1}+Q_{q+1})]_{C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}}

\displaystyle\leq C\lambda_{1}^{(1-2\kappa_{\bar{p}})\gamma(1-\alpha_{0})}\lambda_{q+1}^{-\theta_{V}(1-\alpha_{0})+B\alpha_{0}}+C\lambda_{q+1}^{-\theta_{Q}+\eta\alpha_{0}}.

The displayed powers of $\lambda_{1}$ are retained explicitly. Once the base frequency is fixed they are independent of $q$ ; the first stage is a single finite term, and the strict negative powers of $\lambda_{q+1}$ control the summability of all later stages. Choose $\alpha_{0}>0$ so small that all four exponents above are strictly negative. Since $\delta_{q+1}$ is a fixed multiple of $\lambda_{q+1}^{-\gamma}$ , there is then a $\vartheta>0$ , independent of $q$ , such that the preceding bound and the already established $L_{t}^{\infty}L_{x}^{\bar{p}}$ bound yield

\|\nabla u_{q+1}\|_{C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}}\leq\|\nabla u_{q}\|_{C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}}+C_{\lambda_{1}}\delta_{q+1}^{\vartheta}.

Here $C_{\lambda_{1}}<\infty$ may depend on the fixed base frequency $\lambda_{1}$ , but is independent of $q$ . This dependence is harmless because $\lambda_{1}$ is fixed throughout the iteration.

Finally, (4.47) and (4.25) give the time-derivative estimate

	$\displaystyle\\|\partial_{t}u_{q+1}\\|_{L^{\infty}_{t}L^{p}_{x}}$	$\displaystyle\leq\\|\partial_{t}u_{l}\\|_{L^{\infty}_{t}L^{p}_{x}}+\sup_{i,k}\\|\partial_{t}V_{i}^{k}\\|_{L^{\infty}_{t}L^{p}_{x}}+\\|\partial_{t}Q_{q+1}\\|_{L^{\infty}_{t}L^{p}_{x}}$
		$\displaystyle\leq\lambda_{q}^{n}+C\delta_{q+1}^{-2}r_{q+1}^{-3}(1+r_{q+1}\lambda_{q+1}^{4\frac{n}{\sigma}+4\gamma})+C\lambda_{q+1}^{4\frac{n}{\sigma}+4\gamma+3-\frac{3}{p}}.$

We apply this general estimate with $p=4$ . The bound $\|\partial_{t}u_{q+1}\|_{L_{t}^{\infty}L_{x}^{4}}\leq\lambda_{q+1}^{n}$ therefore requires

4\frac{n}{\sigma}+4\gamma+3-\frac{3}{4}<n,\quad 3\mu+2\gamma<n,\quad 2\mu+2\gamma+4\frac{n}{\sigma}+4\gamma<n.

(5.5)

5.2 Reynolds-stress estimates

We estimate the components of the Reynolds stress defined in the preceding section. Each component will be bounded by a fixed fraction of the target level $\delta_{q+2}$ . We treat successively the linear interaction, the temporal corrector, the freezing and source-replacement errors, and the intrinsic errors of the moving blocks.

By (4.21), Poincaré’s inequality, and Hölder’s inequality,

	$\displaystyle\\|\mathcal{E}^{(l)}_{q+1}\\|_{L^{\infty}_{t}L^{1}_{x}}$	$\displaystyle\leq C\\|v_{q+1}\\|_{L^{\frac{3}{2}}}\\|u_{l}\\|_{L^{3}}\leq C\\|v_{q+1}\\|_{L^{\frac{3}{2}}}\\|\nabla u_{l}\\|_{L^{4}}$
		$\displaystyle\leq C\delta_{q+1}^{\frac{1}{2}}r_{q+1}^{\frac{1}{3}}\left(\lambda_{q+1}^{\frac{n}{\sigma}+\gamma}\right)\lambda_{q}^{n}.$

When this and the other terms containing $\delta_{q+1}^{1/2}$ are compared with $\delta_{q+2}$ , we use the exact ratio

\frac{\delta_{q+1}^{1/2}}{\delta_{q+2}}=\lambda_{1}^{-\gamma}\lambda_{q+1}^{\gamma\sigma-\gamma/2}.

Thus the retained factor $\lambda_{1}^{-\gamma}$ is favorable, and the displayed exponent inequalities remain sufficient. After choosing $\lambda_{0}$ sufficiently large to absorb the fixed constant, the bound $\|\mathcal{E}^{(l)}_{q+1}\|_{L_{t}^{\infty}L_{x}^{1}}\leq\delta_{q+2}/100$ follows from the exponent inequality

\lambda_{q+1}^{2\frac{n}{\sigma}+\gamma-\frac{\mu}{3}-\frac{\gamma}{2}}<\lambda_{q+1}^{-\gamma\sigma}.

(5.6)

For the corrector stress, we estimate

	$\displaystyle\\|\mathcal{E}_{q+1}^{(c)}\\|_{L_{t}^{\infty}L_{x}^{1}}$	$\displaystyle\leq C\\|Q_{q+1}\\|_{L_{t}^{\infty}L_{x}^{2}}\bigl(\\|u_{l}\\|_{L_{t}^{\infty}L_{x}^{2}}+\\|v_{q+1}\\|_{L_{t}^{\infty}L_{x}^{2}}+\\|Q_{q+1}\\|_{L_{t}^{\infty}L_{x}^{2}}\bigr)+C\\|\nabla Q_{q+1}\\|_{L_{t}^{\infty}L_{x}^{6/5}}$
		$\displaystyle\leq C\tau_{q+1}\lambda_{q+1}^{\frac{3}{2}+4\frac{n}{\sigma}+4\gamma}\delta_{0}^{\frac{1}{2}}{\leq\frac{1}{100}\delta_{q+2}}.$

It is therefore sufficient that

\lambda_{q+1}^{4\frac{n}{\sigma}+4\gamma+\frac{3}{2}-\eta}<\lambda_{q+1}^{-\gamma\sigma}.

(5.7)

For the coefficient-freezing error,

\displaystyle\|\mathcal{E}_{q+1}^{(t)}\|_{L_{t}^{\infty}L_{x}^{1}}

\displaystyle\leq C\tau_{q+1}\|a_{i}\|_{\dot{C}^{1}}\leq C\tau_{q+1}\lambda_{q+1}^{4\frac{n}{\sigma}+4\gamma}{\leq\frac{1}{100}\delta_{q+2}},

where the last inequality follows from (5.7).

Choose an exponent $p_{s}>1$ for the source-replacement estimate, independently of the exponent used in Proposition 3.1. From Proposition 3.1, (4.36), and (4.30),

	$\displaystyle\\|\mathcal{E}_{q+1}^{(s)}\\|_{L^{\infty}_{t}L^{p_{s}}_{x}}$	$\displaystyle\leq C(p_{s})\lambda_{q+1}^{-1}\left(\\|\frac{1}{R_{i}^{k}}\tilde{H}_{R_{i}^{k}}\\|_{L^{\infty}_{t}L^{p_{s}}_{x}}+\\|\tilde{U}_{i}^{k}\\|_{L^{\infty}_{t}L^{p_{s}}_{x}}\right)\sup_{i,k}\Big\|\frac{\mathrm{d}}{\mathrm{d}t}(\eta_{i}^{k}\zeta_{i}^{k}R_{i}^{k})\Big\|$
		$\displaystyle\leq C(p_{s})\lambda_{q+1}^{-1}(\delta_{q+1}^{\frac{2}{p_{s}}-2}r_{q+1}^{{\frac{2}{p_{s}}-2}}+\lambda_{q+1}^{3-\frac{3}{p_{s}}})\lambda^{4\frac{n}{\sigma}+4\gamma}_{q+1}{\leq\frac{\delta_{q+2}}{100}}.$

Since $L^{p_{s}}(\mathbb{T}^{3})\hookrightarrow L^{1}(\mathbb{T}^{3})$ , the preceding bound controls the required $L^{1}$ stress. It closes provided

\displaystyle-1+\frac{4n}{\sigma}+4\gamma+(\mu+\gamma)\left(2-\frac{2}{p_{s}}\right)

\displaystyle<-\gamma\sigma,

\displaystyle-1+\frac{4n}{\sigma}+4\gamma+3-\frac{3}{p_{s}}

\displaystyle<-\gamma\sigma.

(5.8)

Both inequalities hold by taking $p_{s}>1$ sufficiently close to one. It remains to use (4.20) and (4.34) for the building-block and averaging errors:

	$\displaystyle\\|F_{i}^{k}\\|_{L_{t}^{\infty}L_{x}^{1}}$	$\displaystyle\leq C\delta_{q+1}\,r_{q+1}^{\kappa}\,(\lambda_{q+1}^{3\frac{n}{\sigma}+3\gamma})^{\kappa}\left(\lambda_{q+1}^{4n/\sigma+5\gamma}\right)+C\nu\delta^{\frac{1}{2}}_{q+1}r_{q+1}^{\frac{2}{p_{0}}-\frac{5}{3}}(\lambda_{q+1}^{3\frac{n}{\sigma}+3\gamma})^{\frac{2}{p_{0}}-\frac{5}{3}}\leq\frac{\delta_{q+2}}{100},$		(5.9)
	$\displaystyle\\|G_{i}^{k}\\|_{L_{x}^{1}}$	$\displaystyle\leq C\frac{\\|a_{i}^{k}\\|_{C^{1}}}{\lambda_{q+1}}\leq C\lambda_{q+1}^{4\frac{n}{\sigma}+4\gamma-1}\leq\frac{\delta_{q+2}}{100}\,.$

At a fixed time there are four global error tensors, at most one active tensor $F_{i}^{k}$ , and nine static tensors $G_{i}^{k}$ on the current time cell. Consequently, after increasing $\lambda_{0}$ once more to absorb all fixed constants,

\|\mathcal{E}_{q+1}\|_{L_{t}^{\infty}L_{x}^{1}}\leq\frac{14}{100}\,\delta_{q+2}<\delta_{q+2}.

5.3 Choice of parameters

We collect the restrictions obtained above and exhibit one choice for which they are all strict. We distinguish the intrinsic block exponent $p_{0}$ , the source-replacement exponent $p_{s}$ , and the final regularity exponent $\bar{p}$ , since these exponents enter different estimates. The geometric condition (4.13) gives

1-\alpha\mu+3\alpha\left(\frac{n}{\sigma}+\gamma\right)<0.

For (4.16), the fixed prefactor in $\delta_{q+1}^{1/2}=\lambda_{1}^{\gamma}\lambda_{q+1}^{-\gamma/2}$ must also be retained. Indeed,

\frac{\lambda_{q+1}^{3}r_{q+1}\delta_{q+1}^{1/2}}{\tau_{q+1}}=\lambda_{1}^{\gamma}\lambda_{q+1}^{3-\mu-\gamma/2+\eta}.

Thus the stronger first-stage condition

3-\mu+\frac{\gamma}{2}<-\eta

is sufficient uniformly for every $q\geq 0$ after increasing $\lambda_{1}$ . We next collect (5.1), (5.2),(5.4), (5.5)–(5.9). Since $\sigma$ is a positive integer, (5.7) implies (5.1) and (5.2). The remaining restrictions reduce to

	$\displaystyle 4\frac{n}{\sigma}+4\gamma+3-\frac{3}{4}<n,\quad 3\mu+2\gamma<n,\quad 2\mu+2\gamma+4\frac{n}{\sigma}+4\gamma<n,$
	$\displaystyle 2\frac{n}{\sigma}-\frac{\mu}{3}+\frac{\gamma}{2}<-\gamma\sigma,\quad 4\frac{n}{\sigma}+4\gamma+\frac{3}{2}-\eta<-\gamma\sigma,\quad 4\frac{n}{\sigma}+4\gamma-1<-\gamma\sigma,$
	$\displaystyle-\mu\kappa+(3\frac{n}{\sigma}+3\gamma)\kappa+4\frac{n}{\sigma}+4\gamma<-\gamma\sigma,\quad-\frac{\gamma}{2}+(3\frac{n}{\sigma}+3\gamma-\mu)(\frac{2}{p_{0}}-\frac{5}{3})<-\gamma\sigma,$
	$\displaystyle-\frac{\gamma}{2}+\frac{7}{6}(\mu+\gamma)<n,\quad-\eta+\frac{13}{4}+4\frac{n}{\sigma}+4\gamma<n.$

These conditions are supplemented by the two source-exponent inequalities in (5.8). We first choose $\sigma=200$ , $n=20$ , $\gamma=\frac{1}{1000}$ , $\mu=6+2\gamma=6.002$ , and $\eta=3$ . For the intrinsic block estimates, set $p_{0}=\frac{28}{27}$ and $\alpha=\frac{2}{11}$ . Then

\kappa=\min\left\{\frac{2+3\alpha}{2p_{0}}-4\alpha,2-6\alpha+\frac{3\alpha}{p_{0}},\frac{2}{p_{0}}-\frac{4}{3}\right\}=\frac{1}{2},

and we choose $\beta=\frac{14}{3}+\frac{3\alpha-2}{p_{0}}-6\alpha=\frac{502}{231}$ . Independently, take $p_{s}=\frac{101}{100}$ in (5.8). The two geometric left-hand sides are approximately $-0.03618$ and $-3.0015<-\eta$ . Moreover,

\lambda_{1}^{\gamma}\lambda_{q+1}^{3-\mu-\gamma/2+\eta}=\lambda_{1}^{\gamma}\lambda_{q+1}^{-5\gamma/2}\leq\lambda_{1}^{-3\gamma/2},

which verifies (4.16), also for $q=0$ , once $\lambda_{1}$ is sufficiently large. The inequalities that become less favorable when $\mu$ is increased have the values

3\mu+2\gamma=18.008<20,\qquad 2\mu+2\gamma+4\frac{n}{\sigma}+4\gamma=12.410<20,\qquad-\frac{\gamma}{2}+\frac{7}{6}(\mu+\gamma)=7.003<20.

The two source-exponent left-hand sides in (5.8) are approximately $-0.47713$ and $-0.56630$ , both below $-\gamma\sigma=-0.2$ . The bounds in (5.6), (5.7), and the $G_{i}^{k}$ -part of (5.9) have exponents $-1.80017$ , $-1.096$ , and $-0.596$ , respectively. The two $\mu$ -dependent exponents in the $F_{i}^{k}$ -part of (5.9) are $-2.44550$ and $-1.49310$ . Hence all of these exponents are strictly below $-\gamma\sigma=-0.2$ , and direct substitution verifies the remaining strict inequalities as well.

For the final exponent $\bar{p}>6/5$ , the gradient estimates impose the additional restrictions in (5.3), namely

-\frac{\gamma}{2}+\left(\frac{5}{3}-\frac{2}{\bar{p}}\right)(\mu+\gamma)<0,\qquad-\eta+4-\frac{3}{\bar{p}}+4\frac{n}{\sigma}+4\gamma<0.

For $\sigma=200$ , $n=20$ , $\gamma=10^{-3}$ , $\mu=6+2\gamma=6.002$ , and $\eta=3$ , both inequalities are strict when $\bar{p}=6/5+5\times 10^{-5}$ : their left-hand sides are approximately $-8.31424\times 10^{-5}$ and $-1.09590$ , respectively. In particular, the positive interpolation margins remain available, and the corresponding time Hölder exponent may still be chosen sufficiently small.

6 Proof of the main theorem

We first construct a smooth Reynolds solution on the fixed interval $I_{\ast}\supset[0,1]$ that approximates the two prescribed endpoint fields. Consider

\partial_{t}u_{0}+\operatorname{div}(u_{0}\otimes u_{0})-\nu\Delta u_{0}+\nabla p_{0}=\operatorname{div}\mathcal{E}_{0},\qquad\operatorname{div}u_{0}=0,

(6.1)

where $\mathcal{E}_{0}$ is symmetric.

Fix $u^{(0)},u^{(1)}\in L^{2}_{\sigma}(\mathbb{T}^{3})$ , and let $\rho_{\ell}$ be a standard spatial mollifier. Choose $\chi\in C^{\infty}(I_{\ast};[0,1])$ such that

\chi\equiv 1\quad\hbox{on }I_{\ast}\cap(-\infty,1/4],\qquad\chi\equiv 0\quad\hbox{on }I_{\ast}\cap[3/4,\infty).

Let $\mathcal{R}$ be a symmetric inverse-divergence operator on the torus, normalized so that

\operatorname{div}\mathcal{R}f=f\qquad\text{whenever}\qquad\int_{\mathbb{T}^{3}}f(x)\,\mathrm{d}x=0.

Proposition 6.1 (Smooth initialization).

Define

$\displaystyle u_{0}$	$\displaystyle:=\chi(t)(u^{(0)}\rho_{\ell})(x)+(1-\chi(t))(u^{(1)}\rho_{\ell})(x),$	(6.2)
$\displaystyle p_{0}$	$\displaystyle:=0,$
$\displaystyle\mathcal{E}_{0}$	$\displaystyle:=\mathcal{R}(\partial_{t}u_{0}+\operatorname{div}(u_{0}\otimes u_{0})-\nu\Delta u_{0}).$	(6.3)

Then $(u_{0},p_{0},\mathcal{E}_{0})$ is a smooth solution of (6.1) on $\mathbb{T}^{3}\times I_{\ast}$ . Moreover, for every $\varepsilon>0$ , there exists $\ell>0$ such that

\|u_{0}(\cdot,0)-u^{(0)}\|_{L^{2}}+\|u_{0}(\cdot,1)-u^{(1)}\|_{L^{2}}<\frac{\varepsilon}{2}.

(6.4)

Proof.

Spatial convolution preserves both incompressibility and zero mean. Consequently,

\int_{\mathbb{T}^{3}}\partial_{t}u_{0}+\operatorname{div}(u_{0}\otimes u_{0})-\nu\Delta u_{0}\mathrm{d}x=0

for every $t$ . Thus $\mathcal{E}_{0}$ is well defined and symmetric, and (6.2)–(6.3) give (6.1). Finally, $u_{0}(\cdot,0)=u^{(0)}*\rho_{\ell}$ and $u_{0}(\cdot,1)=u^{(1)}*\rho_{\ell}$ , so (6.4) follows by taking $\ell$ sufficiently small. ∎

After $\ell$ and $\chi$ have been fixed, all relevant norms of the initial triple are finite. By increasing $\lambda_{0}$ , we may arrange

\|\mathcal{E}_{0}\|_{L_{t}^{\infty}L_{x}^{1}}\leq\delta_{1},\quad\|u_{0}\|_{L_{t}^{\infty}L_{x}^{2}}\leq\delta_{0}^{\frac{1}{2}},\quad\|(\nabla_{x}u_{0},\partial_{t}u_{0})\|_{L_{t}^{\infty}L_{x}^{4}}\leq\lambda_{0}^{n}.

Thus $(u_{0},p_{0},\mathcal{E}_{0})$ satisfies the induction hypotheses at level $q=0$ .

We use the following compactness statement.

Proposition 6.2 (Compactness and convergence).

Let $(u_{q},p_{q},\mathcal{E}_{q})_{q\geq 0}$ be a sequence of admissible solutions of (4.2), with every $u_{q}$ divergence free and of zero mean. Suppose that, for some $\alpha_{0},\vartheta>0$ ,

$\displaystyle\\|\mathcal{E}_{q}\\|_{L_{t}^{\infty}L_{x}^{1}}$	$\displaystyle\leq\delta_{q+1},$	(6.5)
$\displaystyle\\|u_{q+1}-u_{q}\\|_{C_{t}L_{x}^{2}}$	$\displaystyle\leq C\delta_{q+1}^{1/2},$	(6.6)
$\displaystyle\\|\nabla u_{q+1}\\|_{C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}}$	$\displaystyle\leq\\|\nabla u_{q}\\|_{C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}}+C\delta_{q+1}^{\vartheta},$	(6.7)
$\displaystyle\\|u_{q+1}(\cdot,j)-u_{q}(\cdot,j)\\|_{L^{2}}$	$\displaystyle\leq C\lambda_{q}^{-1/5},\qquad j=0,1,$	(6.8)

where $\lambda_{q+1}=\lambda_{q}^{\sigma}$ with $\sigma>1$ and $\delta_{q}=\lambda_{1}^{2\gamma}\lambda_{q}^{-\gamma}$ with $\gamma>0$ . Then there exists

u\in C([0,1];L^{2}_{\sigma}(\mathbb{T}^{3}))\quad\text{with}\quad\nabla u\in{C^{\alpha_{0}}}([0,1];L^{\bar{p}}(\mathbb{T}^{3}))

such that $u_{q}\to u$ strongly in $C_{t}L_{x}^{2}$ . The field $u$ is a weak solution of the Navier–Stokes equation in the sense that

\displaystyle\int_{0}^{1}\!\!\int_{\mathbb{T}^{3}}\left[u\cdot\partial_{t}\varphi+(u\otimes u):\nabla\varphi+\nu u\cdot\Delta\varphi\right]\mathrm{d}x\mathrm{d}t+\int_{\mathbb{T}^{3}}u(x,0)\cdot\varphi(x,0)\mathrm{d}x=0

(6.9)

for every divergence-free $\varphi\in C_{c}^{\infty}(\mathbb{T}^{3}\times[0,1);\mathbb{R}^{3})$ . If, in addition, (6.4) holds and $C\sum_{q=0}^{\infty}\lambda_{q}^{-1/5}<\varepsilon/2$ , then

\|u(\cdot,j)-u^{(j)}\|_{L^{2}}<\varepsilon,\qquad j=0,1.

Proof.

Since $\lambda_{q}=\lambda_{0}^{\sigma^{q}}$ , the three series

\sum_{q=0}^{\infty}\delta_{q+1}^{1/2},\qquad\sum_{q=0}^{\infty}\delta_{q+1}^{\vartheta},\qquad\sum_{q=0}^{\infty}\lambda_{q}^{-1/5}

(6.10)

converge. It follows from (6.6) that $(u_{q})_{q}$ is Cauchy in $C([0,1];L^{2})$ , so

u_{q}\longrightarrow u\qquad\text{strongly in }C([0,1];L^{2})

(6.11)

for some $u$ in that space. Divergence and spatial mean pass to the strong $L^{2}$ limit, hence $u(t)\in L^{2}_{\sigma}$ for every $t$ .

Iterating (6.7) and using (6.10) gives

\sup_{q\geq 0}\|\nabla u_{q}\|_{C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}}\leq\|\nabla u_{0}\|_{C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}}+C\sum_{q=0}^{\infty}\delta_{q+1}^{\vartheta}=:M<\infty.

(6.12)

Fix $t\in[0,1]$ . Since $\bar{p}>1$ , $L^{\bar{p}}$ is reflexive, and $(\nabla u_{q}(t))_{q}$ is weakly precompact in $L^{\bar{p}}$ . The strong $L^{2}$ convergence implies convergence in distributions, so every weakly convergent subsequence of $\nabla u_{q}(t)$ has the unique limit $\nabla u(t)$ . Hence the full sequence converges weakly, and

u(t)\in W^{1,\bar{p}}(\mathbb{T}^{3}),\qquad\nabla u_{q}(t)\rightharpoonup\nabla u(t)\quad\text{weakly in }L^{\bar{p}}.

For $s,t\in[0,1]$ , weak lower semicontinuity and (6.12) yield

	$\displaystyle\\|\nabla u(t)-\nabla u(s)\\|_{L^{\bar{p}}}$	$\displaystyle\leq\liminf_{q\to\infty}\\|\nabla u_{q}(t)-\nabla u_{q}(s)\\|_{L^{\bar{p}}}$
		$\displaystyle\leq M\|t-s\|^{\alpha_{0}}.$

Thus $\nabla u\in C_{t}^{\alpha_{0}}L_{x}^{\bar{p}}$ .

It remains to pass to the equation. Testing the $q$ th Reynolds system against a divergence-free $\varphi$ gives

		$\displaystyle\int_{0}^{1}\!\!\int_{\mathbb{T}^{3}}\left[u_{q}\cdot\partial_{t}\varphi+(u_{q}\otimes u_{q}):\nabla\varphi+\nu u_{q}\cdot\Delta\varphi\right]\mathrm{d}x\mathrm{d}t+\int_{\mathbb{T}^{3}}u_{q}(x,0)\cdot\varphi(x,0)\mathrm{d}x$
		$\displaystyle\hskip 99.58464pt=\int_{0}^{1}\!\!\int_{\mathbb{T}^{3}}\mathcal{E}_{q}:\nabla\varphi\mathrm{d}x\mathrm{d}t.$		(6.13)

The right-hand side tends to zero by (6.5). Moreover, (6.11) and the uniform $L^{2}$ bound imply

\displaystyle\|u_{q}\otimes u_{q}-u\otimes u\|_{C_{t}L_{x}^{1}}

\displaystyle\leq\bigl(\|u_{q}\|_{C_{t}L_{x}^{2}}+\|u\|_{C_{t}L_{x}^{2}}\bigr)\|u_{q}-u\|_{C_{t}L_{x}^{2}}\longrightarrow 0.

All the remaining terms in (6) pass to the limit directly by (6.11), including the initial trace. We obtain (6.9). A distributional pressure may then be recovered from the periodic de Rham theorem.

Finally, (6.8) and (6.4) give

\|u(\cdot,j)-u^{(j)}\|_{L^{2}}\leq\|u^{(j)}-u_{0}(\cdot,j)\|_{L^{2}}+\|u(\cdot,j)-u_{0}(\cdot,j)\|_{L^{2}}<\frac{\varepsilon}{2}+C\sum_{q=0}^{\infty}\lambda_{q}^{-1/5}<\varepsilon,\qquad j=0,1.

∎

We now complete the proof of Theorem 1.1. Choose $\ell$ by Proposition 6.1, and then choose $\lambda_{0}$ sufficiently large that the initial triple satisfies the induction hypotheses and $C\sum_{q}\lambda_{q}^{-1/5}<\varepsilon/2$ . Repeated application of Proposition 4.1 gives a sequence satisfying the assumptions of Proposition 6.2. Its limit has the required regularity, solves (1.1) in the sense of (1.2), and satisfies the two endpoint estimates.

6.1 Time locality and proof of Theorem 1.2

The endpoint-density statement alone does not imply exact nonuniqueness. The extra ingredient is that one iteration step only uses the Reynolds flow in a short neighborhood of the current time cell. We record this feature in the form needed below. At stage $q$ , write

l_{q}:=c_{0}\lambda_{q+1}^{-n/\sigma-\gamma}

(6.14)

for the space–time mollification scale used in Section 4.

Lemma 6.3 (Time locality of the iteration).

Suppose that two Reynolds triples satisfying the hypotheses of Proposition 4.1, with the same parameters, agree on $\mathbb{T}^{3}\times(I_{\ast}\cap(-\infty,T_{q}])$ . After fixing the same normalization for the pressures, the two applications of Proposition 4.1 may be coupled so that the resulting triples agree on

\mathbb{T}^{3}\times\bigl(I_{\ast}\cap(-\infty,T_{q}-l_{q}-\tau_{q+1}]\bigr),

provided $T_{q}>l_{q}+\tau_{q+1}$ .

Proof.

Since the two input triples agree on $I_{\ast}\cap(-\infty,T_{q}]$ , their mollifications, and hence the coefficients $a_{i}$ , agree on $I_{\ast}\cap(-\infty,T_{q}-l_{q}]$ . Every complete time cell contained in this interval therefore has the same frozen coefficients $a_{i}^{k}$ in the two constructions. On such a cell we choose the same admissible initial points for the trajectories. The amplitudes, radii, trajectories, moving Hill blocks, and auxiliary sources then agree on that cell.

The temporal corrector (4.44) is defined by integration only over the current cell. The Leray projection, the symmetric inverse-divergence operator, and all other operators used to define $u_{q+1}$ , $p_{q+1}$ , and $\mathcal{E}_{q+1}$ act only in space. Consequently, the two output triples agree on the union of the complete cells contained in $I_{\ast}\cap(-\infty,T_{q}-l_{q}]$ . In particular, they agree on $[0,T_{q}-l_{q}-\tau_{q+1}]$ , and the same left-time convention is preserved for the next stage. This proves the claim. ∎

Proof of Theorem 1.2.

Fix an arbitrary $v\in L^{2}_{\sigma}(\mathbb{T}^{3})$ and $\varepsilon>0$ . Choose $w\in L^{2}_{\sigma}(\mathbb{T}^{3})$ with $\|w\|_{L^{2}}=1$ , set

w^{(1)}:=0,\qquad w^{(2)}:=w,\qquad\varepsilon_{*}:=\min\{\varepsilon/2,1/8\},

and choose one spatial mollification scale $\ell>0$ , common to both constructions, such that

\|v*\rho_{\ell}-v\|_{L^{2}}+\sum_{j=1}^{2}\|w^{(j)}*\rho_{\ell}-w^{(j)}\|_{L^{2}}<\frac{\varepsilon_{*}}{2}.

For $j=1,2$ , apply Proposition 6.1 with endpoint targets $(v,w^{(j)})$ , using this common scale and the same cutoff $\chi$ . Denote the resulting initial Reynolds triples by $(u_{0}^{(j)},p_{0}^{(j)},\mathcal{E}_{0}^{(j)})$ . Since $\chi\equiv 1$ on $I_{\ast}\cap(-\infty,1/4]$ , the two triples agree on $\mathbb{T}^{3}\times(I_{\ast}\cap(-\infty,1/4])$ .

Choose one $\lambda_{0}$ , common to both constructions, sufficiently large that both initial triples satisfy the induction hypotheses,

C\sum_{q=0}^{\infty}\lambda_{q}^{-1/5}<\frac{\varepsilon_{*}}{2},\qquad\sum_{q=0}^{\infty}(l_{q}+\tau_{q+1})<\frac{1}{8}.

The second requirement is possible because $l_{q}$ is given by (6.14), $\tau_{q+1}=\lambda_{q+1}^{-\eta}$ , and $\lambda_{q}=\lambda_{0}^{\sigma^{q}}$ . Apply Proposition 4.1 to the two triples with identical choices on every time cell on which they agree. Iterating Lemma 6.3, the level- $q$ triples agree on

\mathbb{T}^{3}\times\bigl(I_{\ast}\cap(-\infty,T_{q}]\bigr),\qquad T_{q}:=\frac{1}{4}-\sum_{m=0}^{q-1}(l_{m}+\tau_{m+1})\geq\frac{1}{8}.

Proposition 6.2 now gives two limiting weak solutions $u^{(1)},u^{(2)}\in\mathcal{S}_{\bar{p}}$ . Their equality on the intervals above and their convergence in $C_{t}L_{x}^{2}$ imply

u^{(1)}(\cdot,t)=u^{(2)}(\cdot,t)\quad\text{for }0\leq t\leq\frac{1}{8}.

In particular, they have a common initial trace, denoted by $\widetilde{v}$ , and the endpoint estimate in Proposition 6.2 gives

\|\widetilde{v}-v\|_{L^{2}}<\varepsilon_{*}\leq\varepsilon.

(6.15)

At the other endpoint,

\|u^{(j)}(\cdot,1)-w^{(j)}\|_{L^{2}}<\varepsilon_{*},\qquad j=1,2.

Therefore,

	$\displaystyle\\|u^{(1)}(\cdot,1)-u^{(2)}(\cdot,1)\\|_{L^{2}}$	$\displaystyle\geq\\|w^{(1)}-w^{(2)}\\|_{L^{2}}-2\varepsilon_{*}$
		$\displaystyle\geq 1-\frac{1}{4}>0,$

so the two solutions are distinct.

Let $\mathcal{D}$ be the set of all initial data in $L^{2}_{\sigma}(\mathbb{T}^{3})$ admitting two such distinct solutions. Since $v$ and $\varepsilon$ were arbitrary, (6.15) shows that $\mathcal{D}$ is dense in $L^{2}_{\sigma}(\mathbb{T}^{3})$ . This proves the theorem. ∎

7 Limitations and possible modifications

We record the limits of the present single-core construction under the condition $\nabla u\in C_{t}L_{x}^{\bar{p}}$ . These calculations are not impossibility results for other Hill-vortex schemes.

7.1 Can the present method improve the exponent?

The principal obstruction is already visible in (5.3). If

\vartheta(p):=\frac{5}{3}-\frac{2}{p},

then the formal principal-gradient estimate would require

\vartheta(\bar{p})<\frac{\gamma}{2(\mu+\gamma)}.

For the parameters used in the construction above, $\gamma=10^{-3}$ and $\mu=6+2\gamma=6.002$ , this gives

\bar{p}<p_{\max}:=\frac{2}{\frac{5}{3}-\frac{\gamma}{2(\mu+\gamma)}}=\frac{24012}{20009}\approx 1.20005997.

(7.1)

Thus $\bar{p}=1.20005=6/5+5\times 10^{-5}$ is already close to the ceiling imposed by the principal-gradient estimate for these parameters.

This value is not claimed to be optimal. However, $\gamma$ and $\mu$ have competing roles: increasing $\gamma$ improves amplitude decay, whereas decreasing $\mu$ enlarges the cores. The orbit-spacing condition

1-\alpha\mu+3\alpha\left(\frac{n}{\sigma}+\gamma\right)<0

keeps $\mu$ relatively large, while the stress estimates restrict $\gamma$ . Reoptimization may improve the numerical margin but does not provide a route toward $p=3/2$ ; a larger gain requires a structural change in the construction.

7.2 Why temporal concentration does not improve the present theorem

Temporal concentration, as used in related convex-integration constructions such as [31, 32], gives estimates in time-integrated spaces. Concentrating a stress of size $\delta$ on a set of relative measure $\rho$ changes the perturbation amplitude to $\delta^{1/2}\rho^{-1/2}$ . Since here $R=ra$ with $a\sim\delta/\rho$ , one obtains

\|\nabla V_{i}^{k}\|_{L_{t}^{s}L_{x}^{p}}\lesssim\rho^{\frac{1}{s}-\frac{1}{2}+\vartheta(p)}\delta^{\frac{1}{2}-\vartheta(p)}r^{-\vartheta(p)}.

For $p$ slightly above $6/5$ , the factor at $s=\infty$ is $\rho^{-1/2+\vartheta(p)}$ , which diverges as $\rho\downarrow 0$ . Temporal cutoffs also worsen the transport and corrector errors. Hence this device does not improve the present uniform-in-time estimate, although it may be useful for a theorem with finite time integrability.

7.3 Anisotropic localization

This direction is motivated by intermittent Beltrami, Mikado, and pipe constructions; see [42, 26, 28, 39, 81, 24, 53]. These schemes use anisotropic spatial concentration. The solutions in [26] belong to $C_{t}(H_{x}^{\beta}\cap W_{x}^{1,1+\beta})$ for some $\beta>0$ , and hence to $C_{t}L_{x}^{2+a}$ for some unspecified $a>0$ . The checkerboard localization of Novack and Vicol [85] gives a related Euler example. This motivated the target

C_{t}\bigl(W_{x}^{1,p}\cap L_{x}^{2+a}\bigr),\qquad p>\frac{6}{5},

with a larger velocity-integrability gain. One may test the cutoff

\chi_{R}^{a,b}(x):=\chi\left(\frac{x_{1}}{R^{a}},\frac{(x_{2},x_{3})}{R^{b}}\right),\qquad 0<a,b<\frac{2}{3},

and set

H_{R}^{A}:=H_{R}-R\nabla(\chi_{R}^{a,b}\Phi).

If $H_{R}^{A}=H_{R}$ on $B_{cR^{2/3}}$ , the unchanged core gives

\|H_{R}^{A}\|_{L^{p}(\mathbb{T}^{3})}\geq\|H_{R}\|_{L^{p}(B_{cR^{2/3}})}\gtrsim R^{\frac{2}{p}-1},\qquad\|\nabla H_{R}^{A}\|_{L^{p}(\mathbb{T}^{3})}\geq\|\nabla H_{R}\|_{L^{p}(B_{cR^{2/3}})}\gtrsim R^{\frac{2}{p}-\frac{5}{3}}.

Thus the cutoff does not change the intrinsic Hill-core loss. It could help only through better packing or a smaller divergence-correction error, neither of which follows from the present estimates.

7.4 Several parallel moving cores

Another possibility is to split one directional stress among $N=N_{q+1}$ disjoint cores moving in parallel. The cancellation normalization would then give an amplitude of order

\eta_{N}\sim\delta_{q+1}^{1/2}N^{-1/2}

for each core. If the cores have comparable radius $R$ and disjoint supports, their combined fixed-time gradient satisfies schematically

\left\|\sum_{j=1}^{N}\nabla V_{j}\right\|_{L^{p}}\lesssim\delta_{q+1}^{1/2}N^{\frac{1}{p}-\frac{1}{2}}R^{\frac{2}{p}-\frac{5}{3}}.

For $p<2$ , the factor $N^{1/p-1/2}$ grows with $N$ . Thus splitting the amplitude does not improve the fixed-time gradient estimate unless the new packing permits sufficiently larger cores. A multicore improvement therefore requires a different packing or averaging mechanism, not only more copies of the same block.

8 Further directions and open problems

We conclude with several questions that lie beyond the present single-core iteration.

Optimal uniform-in-time spatial regularity.

Determine the largest exponent $p$ for which the endpoint-trace density in (1.3) can hold with $\nabla u\in C_{t}L_{x}^{p}$ . The ceiling in (7.1) concerns only the chosen parameters. Can optimization, localization, or a different averaging mechanism give a fixed gain above $6/5$ ? The scale-invariant value is $p=3/2$ , for which $W^{1,3/2}(\mathbb{T}^{3})\hookrightarrow L^{3}(\mathbb{T}^{3})$ , but the present estimates do not approach it.

Energy profiles and admissibility.

Prescribed energy profiles and admissibility criteria have been studied in Euler convex integration; see [41, 38, 65, 23]. Can the construction prescribe $E(t)=\frac{1}{2}\|u(\cdot,t)\|_{L^{2}}^{2}$ while retaining $\nabla u\in C_{t}L_{x}^{\bar{p}}$ , or give flexibility among energy-compatible endpoints in the Leray–Hopf or suitable class? Arbitrary endpoint density is incompatible with nonincreasing energy, and the present estimates give neither $L_{t}^{2}H_{x}^{1}$ control nor a local energy inequality.

Coherent vortices beyond the Hill profile.

The Hill energy–gradient scaling gives the threshold $6/5$ . Possible alternatives include Fraenkel’s rings of small cross-section [48], Norbury’s one-parameter family [84] interpolating between thin rings and Hill’s vortex, and the variational constructions of Fraenkel and Berger [49] and Friedman and Turkington [50]. Their aspect ratio provides an additional parameter, although the induced velocity may offset the apparent $L^{p}$ -gain. Hill’s vortex is also the unique extreme member of the family [6]. Any replacement must retain exact traveling cancellation, a controllable impulse [91, Ch. 3], localizability, and small viscous and modulation stresses. Whether a vortex-ring profile can meet these requirements and improve the exponent is open.

Numerically assisted search for new building blocks.

Numerical and computer-assisted work on Navier–Stokes nonuniqueness [62, 63] studies candidate self-similar profiles and unstable modes; related tools appear for Euler [30, 102] and in rigorous a posteriori error estimates [11]. For the present method, one would instead seek a traveling or pulsating profile with better energy–gradient scaling and a certified residual small in the required stress norms. Numerical optimization may help identify candidates, but the cited instability mechanisms do not supply the cancellation identities needed here.

Trajectory geometry and nonperiodic domains.

Extending the construction to $\mathbb{R}^{3}$ requires a substitute for periodic recurrence and control of spatial decay; bounded domains also require boundary-compatible blocks. For three-dimensional Euler, Enciso, Peñafiel-Tomás, and Peralta-Salas [45] proved that, under explicit compatibility conditions, a smooth local solution on a bounded region can be extended to an admissible $C^{\beta}$ weak solution on $\mathbb{R}^{3}$ , $\beta<1/3$ , with controlled support. Their result does not provide boundary-compatible Hill vortices or treat viscosity. The inner–outer gluing in [3] is another precedent. On the torus, partial transport by the coarse velocity may reduce the linear interaction, but would require averaging along deformed trajectories.

These questions ask whether the gain above $6/5$ extends beyond the present moving-Hill-vortex construction.

Acknowledgment

Quoc-Hung Nguyen’s research was supported by the CAS Project for Young Scientists in Basic Research (Grant No. YSBR-031) and by the National Natural Science Foundation of China under Grant Nos. 1251101538 and 12595282. He is grateful to Elia Bruè for suggesting that he consider this problem.

AI use disclosure

This paper was written by the authors and was not generated by artificial intelligence. OpenAI’s ChatGPT 5.6 was used to assist with language editing, organization, bibliographic checks, LaTeX troubleshooting, and verification of mathematical calculations. It was not used as an independent source of mathematical results. All mathematical ideas, arguments, proofs, and conclusions are those of the authors, who reviewed the AI-assisted revisions and take full responsibility for the contents of the paper.

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	$\displaystyle\\|\operatorname{div}(\bar{H}_{R}\circledast K_{R})\\|_{L^{\infty}}$	$\displaystyle\leq C\\|K_{R}\\|_{L^{1}}\\|\nabla\bar{H}_{R}\\|_{L^{\infty}}\leq CR^{\beta-\frac{8}{3}}\leq CR^{-2},$
	$\displaystyle\\|\nabla\operatorname{div}(\bar{H}_{R}\circledast K_{R})\\|_{L^{\infty}}$	$\displaystyle\leq C\\|\nabla K_{R}\\|_{L^{1}}\\|\nabla\bar{H}_{R}\\|_{L^{\infty}}\leq CR^{-8/3}.$

	$\displaystyle\\|V\\|_{L_{t}^{\infty}L_{x}^{q}}$	$\displaystyle\leq C\\|\eta\\|_{\infty}\\|R^{\frac{2}{q}-1}\\|_{L^{\infty}(I)},$
	$\displaystyle\\|\nabla V\\|_{L_{t}^{\infty}L_{x}^{q}}$	$\displaystyle\leq C\\|\eta\\|_{\infty}\\|R^{\frac{2}{q}-\frac{5}{3}}\\|_{L^{\infty}(I)},$
	$\displaystyle\\|\partial_{t}V\\|_{L_{t}^{\infty}L_{x}^{q}}$	$\displaystyle\leq C\\|\eta\\|_{\infty}^{2}\\|R^{-3}\\|_{L^{\infty}(I)}(1+\\|\nabla\mathfrak{R}\\|_{\infty})+C\\|\eta^{\prime}\\|_{\infty}\\|R^{-1}\\|_{L^{\infty}(I)},$
	$\displaystyle\\|\partial_{t}\nabla V\\|_{L_{t}^{\infty}L_{x}^{q}}$	$\displaystyle\leq C\\|\eta\\|_{\infty}^{2}\\|R^{-3-\beta}\\|_{L^{\infty}(I)}(1+\\|\nabla\mathfrak{R}\\|_{\infty})+C\\|\eta^{\prime}\\|_{\infty}\\|R^{-\frac{5}{3}}\\|_{L^{\infty}(I)}.$

	$\displaystyle\\|F_{3}\\|_{L_{t}^{\infty}L_{x}^{1}}$	$\displaystyle\leq C\\|F_{3}\\|_{L_{t}^{\infty}L_{x}^{p}}$
		$\displaystyle\leq C\\|\eta\\|_{L^{\infty}}^{2}\left(\\|R^{2-6\alpha+\frac{3\alpha}{p}}\\|_{L^{\infty}(I)}+\\|\nabla\log\mathfrak{R}\\|_{L^{\infty}}\\|R^{\frac{2}{p}-\frac{4}{3}}\\|_{L^{\infty}(I)}\right).$		(3.8)

	$\displaystyle\\|V^{c}\\|_{L^{q}}$	$\displaystyle\leq C\\|V^{p}\\|_{L^{q}},$	$\displaystyle\\|\nabla V^{c}\\|_{L^{q}}$	$\displaystyle\leq C\\|\nabla V^{p}\\|_{L^{q}},$
	$\displaystyle\\|\partial_{t}V^{c}\\|_{L^{q}}$	$\displaystyle\leq C\\|\partial_{t}V^{p}\\|_{L^{q}},$	$\displaystyle\\|\partial_{t}\nabla V^{c}\\|_{L^{q}}$	$\displaystyle\leq C\\|\partial_{t}\nabla V^{p}\\|_{L^{q}}.$

	$\displaystyle\\|\partial_{t}V^{p}\\|_{L^{\infty}}$	$\displaystyle\leq\frac{\|\eta\|^{2}}{R}\\|\nabla\widetilde{H}_{R,e}\\|_{L^{\infty}}+\|\eta^{\prime}\|\\|\widetilde{H}_{R,e}\\|_{L^{\infty}}$
		$\displaystyle\quad+\|\eta R^{\prime}\|\left(\frac{1}{R}\\|\widetilde{H}_{R,e}\\|_{L^{\infty}}+\\|\nabla\widetilde{P}_{2}\\|_{L^{\infty}}+\\|\operatorname{div}\widetilde{F}_{2}\\|_{L^{\infty}}\right)$
		$\displaystyle\leq C\left(\|\eta\|^{2}R^{-\frac{8}{3}}+\|\eta^{\prime}\|R^{-1}+\|\eta R^{\prime}\|R^{-2}\right),$

Flexibility for the Three-Dimensional Navier–Stokes Equations via Moving Hill Vortices

Abstract

1 Introduction

1.1 The problem and the main result

Theorem 1.1 (Flexibility).

1.2 Interpretation and consequences of the theorem

Theorem 1.2 (Nonuniqueness).

1.3 Background and related work

1.4 Moving Hill vortices and the critical scaling

The Hill profile.

1.5 Strategy of the construction

The zeroth Reynolds flow and the iteration

Stress decomposition and moving Hill blocks

Reconstruction of the stress by time averages

The temporal corrector and the new stress

1.6 Crossing the Hill threshold and analytical difficulties

Competing effects of concentration

The balance above 6/56/5

Localization, background interaction, and time continuity

1.7 Organization of the paper

2 Hill’s spherical vortex

Remark 2.1.

2.1 Scaled Hill vortex profile

Remark 2.2.

2.2 Estimates for the building block

Lemma 2.3.

Proof.

Proposition 2.4.

Proof.

Corollary 2.5 (Mollified Hill block).

Proof.

3 The moving Hill block

Proposition 3.1 (Moving Hill block).

Proof.

Corollary 3.2 (Navier–Stokes moving Hill block).

Proof.

4 The iteration step

Proposition 4.1.

4.1 Mollification step

4.2 Error decomposition

Lemma 4.2 (Error Decomposition).

Proof.

4.3 Time partition and cutoff functions

4.4 Space-dependent vortex scale

4.5 Trajectory of the vortex core

4.6 Estimates of the building block

4.7 Auxiliary building block

Proposition 4.3.

Proof of Proposition 4.3.

4.8 The temporal corrector

4.9 Construction of the next Reynolds flow

Remark 4.4.

5 Proof of Proposition 4.1

5.1 Velocity estimates

Remark 5.1.

Remark 5.2.

5.2 Reynolds-stress estimates

5.3 Choice of parameters

6 Proof of the main theorem

Proposition 6.1 (Smooth initialization).

Proof.

Proposition 6.2 (Compactness and convergence).

Proof.

6.1 Time locality and proof of Theorem 1.2

Lemma 6.3 (Time locality of the iteration).

Proof.

Proof of Theorem 1.2.

7 Limitations and possible modifications

7.1 Can the present method improve the exponent?

7.2 Why temporal concentration does not improve the present theorem

7.3 Anisotropic localization

7.4 Several parallel moving cores

8 Further directions and open problems

Optimal uniform-in-time spatial regularity.

Energy profiles and admissibility.

Coherent vortices beyond the Hill profile.

Numerically assisted search for new building blocks.

Trajectory geometry and nonperiodic domains.

Acknowledgment

AI use disclosure

The balance above $6/5$