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 \({q\neq\infty}\). 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 et al. (Uspekhi Mat Nauk 58:3–44, 2003; translation in Russ Math Surv 58, 211–250, 2003), 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 \({L_t^{\infty}(L_x^{3,q})}\), \({q\neq\infty}\), is also obtained as a consequence.