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 Luloc,σq(+d). We prove the analyticity of the Stokes semigroup etA in Luloc,σq(+d) for 1 < q . This follows from the analysis of the Stokes resolvent problem for data in Luloc,σq(+d), 1 < q . 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.

Keywords
Navier–Stokes equations, resolvent estimates, analyticity, Stokes semigroup, local uniform Lebesgue spaces, mild solutions, concentration, Liouville theorems, pressure, half-space
Mathematical Subject Classification 2010
Primary: 35A01, 35A02, 35B44, 35B53, 35Q30
Secondary: 35C99, 76D03, 76D05
Milestones
Received: 16 November 2017
Revised: 4 March 2019
Accepted: 18 April 2019
Published: 13 June 2020
Authors
Yasunori Maekawa
Department of Mathematics
Kyoto University
Kyoto
Japan
Hideyuki Miura
Department of Mathematical and Computing Sciences
Tokyo Institute of Technology
Tokyo
Japan
Christophe Prange
CNRS, UMR 5251, IMB
Université de Bordeaux
Bordeaux
France