On the integrability properties of Leray-Hopf solutions of the Navier-Stokes equations on $\mathbb{R}^3$
Sauli Lindberg
TL;DR
The paper investigates how Leray-Hopf solutions to the 3D Navier–Stokes equations on $\mathbb{R}^3$ interact with integrability conditions in spaces $L^r(0,\infty;X)$ and related Besov spaces, and how these relate to uniform-in-viscosity bounds and energy equality. Using a Baire-category framework inspired by Guerra–Koch–Lindberg (GKL23) and a relaxation approach for the Euler equations, the authors establish that certain supercritical integrability classes are generically impossible for Leray-Hopf solutions, while in critical regimes solvability is equivalent to specific a priori estimates. They derive Besov-space regularity consequences, connect time-weighted LPS classes to global regularity criteria, and show that many integrability and energy-equality questions reduce to the presence of uniform-in-viscosity bounds and to the validity of Euler-limit estimates. The results imply that generic data fail to satisfy $L^4(0,T;L^4)$ or related higher integrability, clarify the role of Besov regularity in energy conservation, and yield necessary conditions for global regularity and Euler dissipation, with extensions to the torus. These findings illuminate the delicate balance between nonlinearity, dissipation, and scaling in high-Reynolds-number regimes and inform Onsager-type questions about energy conservation in inviscid limits.
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 $u_0 \in L^2_σ$ 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 $u_0 \in L^2_σ$, no weak solution $u^ν$ 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 $\|u^ν\|_{L^r(0,\infty;X)} \leq C ν^{3-5/r-6/s} \|u_0\|_{L^2}^{4/r+6/s-2}$. Furthermore, we can only have $\limsup_{ν\to 0} \|u^ν\|_{L^r(0,\infty;Z)} < \infty$ for all $u_0 \in L^2_σ$ 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 $L^2_σ$ 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 $Z \hookrightarrow L^2_σ$, 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 $L^2_σ$ 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 $L^2_σ$ datum. These two results also hold on the torus $\mathbb{T}^3$.