The Navier-Stokes equations in nonendpoint borderline 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 $q\ne \infty$. Here $L_x^{3,q}$, $0<q\le \infty$, is an increasing scale of Lorentz spaces containing $L_x^3$. Thus the result provides an improvement of a result by Escauriaza, Seregin and {\v S}verák ((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 $L_t^{\infty }(L_x^{3,q})$, $q\ne \infty$, is also obtained as a consequence.

Minor revision with references added