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

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 $X\subset {\mathcal{S}}^{\prime }$: does every $u_0\in L_{\sigma }^2$ 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 $u_0\in L_{\sigma }^2$, no weak solution $u^{\nu }$ belongs to $L^r(0,\infty ;X)$. If $\frac{3}{2}-\frac{1}{2r}\le \frac{2}{r}+\frac{3}{s}<\frac{3}{2}$ instead, global solvability in $L^r(0,\infty ;X)$ is equivalent to the a priori estimate $\Vert u^{\nu }{\Vert }_{L^r(0,\infty ;X)}\le C{\nu }^{3-5/r-6/s}\Vert u_0{\Vert }_{L^2}^{4/r+6/s-2}$. Furthermore, we can only have ${\mathrm{limsup}}_{\nu \to 0}\Vert u^{\nu }{\Vert }_{L^r(0,\infty ;Z)}<\infty$ for all $u_0\in L_{\sigma }^2$ 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 $L_{\sigma }^2$ 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 $Z\hookrightarrow L_{\sigma }^2$, 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 $\Vert u{\Vert }_{L^3(0,\infty ;{\dot{B}}_{3,\infty }^{1/3})}\le C\Vert u_0{\Vert }_Z^{2/3}$ holds. As a by-product we show that if global regularity holds for the Navier-Stokes equations, then for a Baire generic $L_{\sigma }^2$ 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 $L_{\sigma }^2$ datum. These two results also hold on the torus ${\mathbb{T}}^3$.

Inaccuracies in the statement and proof of Theorem 1.2 corrected, other results unchanged