The Navier-Stokes equations in the critical Lebesgue space

TL;DR

This paper establishes regularity and uniqueness of solutions to the Navier-Stokes equations under certain Lebesgue space conditions, extending previous results and analyzing long-term behavior of solutions.

Contribution

It proves that solutions in a critical Lebesgue space are smooth and unique, and shows decay of solutions over infinite time, generalizing prior regularity criteria.

Findings

Solutions in L_infinity^t L_d^x are smooth and unique.

Solutions decay to zero as time approaches infinity.

Generalizes previous regularity results for Navier-Stokes equations.

Abstract

We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove in this paper that if $u\in L_\infty^tL_{d}^x((0,T)\times {\mathbb R}^d)$ is a Leray-Hopf weak solution, then $u$ is smooth and unique in $(0,T)\times \bR^d$. This generalizes a result by Escauriaza, Seregin and \v{S}ver\'ak. Additionally, we show that if $T=\infty$ then $u$ goes to zero as $t$ goes to infinity.