Vol. 13, No. 4, 2020

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

Yasunori Maekawa, Hideyuki Miura and Christophe Prange

Vol. 13 (2020), No. 4, 945–1010
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. This follows from the analysis of the Stokes resolvent problem for data in ${L}_{uloc,\sigma }^{q}\left({ℝ}_{+}^{d}\right)$, $1. 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.

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