On the integrability properties of Leray-Hopf solutions of the Navier-Stokes equations on $\mathbb{R}^3$

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Abstract

Let $r,s \in [2,\infty]$ and consider the Navier-Stokes equations on $\mathbb{R}^3$. We study the following two questions for suitable $s$-homogeneous Banach spaces $X \subset \mathcal{S}'$: does every $u_0 \in L^2_\sigma$ have a weak solution that belongs to $L^r(0,\infty;X)$, and are the $L^r(0,\infty;X)$ norms of the solutions bounded uniformly in viscosity? We show that if $\frac{2}{r} + \frac{3}{s} < \frac{3}{2}-\frac{1}{2r}$, then for a Baire generic datum $u_0 \in L^2_\sigma$, no weak solution $u^\nu$ belongs to $L^r(0,\infty;X)$. If $\frac{3}{2}-\frac{1}{2r} \leq \frac{2}{r} + \frac{3}{s} < \frac{3}{2}$ instead, global solvability in $L^r(0,\infty;X)$ is equivalent to the a priori estimate $\|u^\nu\|_{L^r(0,\infty;X)} \leq C \nu^{3-5/r-6/s} \|u_0\|_{L^2}^{4/r+6/s-2}$. Furthermore, we can only have $\limsup_{\nu \to 0} \|u^\nu\|_{L^r(0,\infty;Z)} < \infty$ for all $u_0 \in L^2_\sigma$ if $\frac{2}{r} + \frac{3}{s}= \frac{3}{2}-\frac{1}{2r}$. The above results and their variants rule out, for a Baire generic $L^2_\sigma$ datum, $L^4(0,T;L^4)$ integrability and various other known sufficient conditions for the energy equality. As another application, for suitable 2-homogeneous Banach spaces $Z \hookrightarrow L^2_\sigma$, each $u_0 \in Z$ has a Leray-Hopf solution $u \in L^3(0,\infty;\dot{B}_{3,\infty}^{1/3})$ if and only if a uniform-in-viscosity bound $\|u\|_{L^3(0,\infty;\dot{B}_{3,\infty}^{1/3})} \leq C \|u_0\|_Z^{2/3}$ holds. As a by-product we show that if global regularity holds for the Navier-Stokes equations, then for a Baire generic $L^2_\sigma$ datum, the Leray-Hopf solution is unique and satisfies the energy equality. We also show that if global regularity holds in the Euler equations, then anomalous energy dissipation must fail for a Baire generic $L^2_\sigma$ datum. These two results also hold on the torus $\mathbb{T}^3$.