Estimates for the Navier–Stokes equations in the half-space for nonlocalized data

Abstract

This paper is devoted to the study of the Stokes and Navier–Stokes equations, in a half-space, for initial data in a class of locally uniform Lebesgue integrable functions, namely $L_{uloc,\sigma }^q\left({ℝ}_{+}^d\right)$. We prove the analyticity of the Stokes semigroup $e^{-tA}$ in $L_{uloc,\sigma }^q\left({ℝ}_{+}^d\right)$ for $1<q\le \infty$. This follows from the analysis of the Stokes resolvent problem for data in $L_{uloc,\sigma }^q\left({ℝ}_{+}^d\right)$, $1<q\le \infty$. We then prove bilinear estimates for the Oseen kernel, which enables us to prove the existence of mild solutions. The three main original aspects of our contribution are: the proof of Liouville theorems for the resolvent problem and the time-dependent Stokes system under weak integrability conditions, the proof of pressure estimates in the half-space, and the proof of a concentration result for blow-up solutions of the Navier–Stokes equations. This concentration result improves a recent result by Li, Ozawa and Wang and provides a new proof.