The Navier-Stokes equations in nonendpoint borderline Lorentz spaces

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.