Estimates for the Navier-Stokes equations in the half-space for non localized data

TL;DR

This paper advances the understanding of Navier-Stokes equations in a half-space by establishing analyticity, bilinear estimates, and blow-up solution concentration under weak integrability conditions.

Contribution

It introduces new Liouville theorems, pressure estimates, and a concentration result for blow-up solutions, enhancing prior knowledge of Navier-Stokes behavior in non-localized data settings.

Findings

Proved analyticity of the Stokes semigroup in $L^q_{uloc,\sigma}$.

Established bilinear estimates for the Oseen kernel.

Provided a new concentration result for blow-up solutions.

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 $L^q_{uloc,\sigma}(\R^d_+)$. We prove the analyticity of the Stokes semigroup $e^{-t{\bf A}}$ in $L^q_{uloc,\sigma}(\R^d_+)$ for $1<q\leq\infty$. This follows from the analysis of the Stokes resolvent problem for data in $L^q_{uloc,\sigma}(\R^d_+)$, $1<q\leq\infty$. We then prove bilinear estimates for the Oseen kernel, which enables to prove the existence of mild solutions. The three main original aspects of our contribution are: (i) the proof of Liouville theorems for the resolvent problem and the time dependent Stokes system under weak integrability conditions, (ii) the proof of pressure estimates in the half-space and (iii) 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.