The Navier-Stokes equations in nonendpoint borderline Lorentz spaces

TL;DR

This paper proves regularity of solutions to the 3D Navier-Stokes equations in certain Lorentz spaces, extending previous results and introducing new energy bounds and regularity criteria.

Contribution

It extends regularity results for Navier-Stokes solutions in Lorentz spaces beyond the case q=3, using new energy bounds and backward uniqueness techniques.

Findings

Solutions in L_t^{}(L_x^{3,q}) are regular for qnot=.

Established weak-strong uniqueness of Leray-Hopf solutions in these Lorentz spaces.

Extended previous regularity results to a broader class of Lorentz spaces.

Abstract

It is shown both locally and globally that $L_t^{\infty}(L_x^{3,q})$ solutions to the three-dimensional Navier-Stokes equations are regular provided $q\not=\infty$. Here $L_x^{3,q}$, $0<q\leq\infty$, is an increasing scale of Lorentz spaces containing $L^3_x$. Thus the result provides an improvement of a result by Escauriaza, Seregin and {\v S}ver\'ak ((Russian) Uspekhi Mat. Nauk {\bf 58} (2003), 3--44; translation in Russian Math. Surveys {\bf 58} (2003), 211--250), which treated the case $q=3$. A new local energy bound and a new $\epsilon$-regularity criterion are combined with the backward uniqueness theory of parabolic equations to obtain the result. A weak-strong uniqueness of Leray-Hopf weak solutions in $L_t^{\infty}(L_x^{3,q})$, $q\not=\infty$, is also obtained as a consequence.