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

Maekawa, Yasunori; Miura, Hideyuki; Prange, Christophe

doi:10.2140/apde.2020.13.945

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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

DOI: 10.2140/apde.2020.13.945

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 $L_{uloc, σ}^{q} (ℝ_{+}^{d})$ . We prove the analyticity of the Stokes semigroup $e^{- t A}$ in $L_{uloc, σ}^{q} (ℝ_{+}^{d})$ for $1 < q \leq \infty$ . This follows from the analysis of the Stokes resolvent problem for data in $L_{uloc, σ}^{q} (ℝ_{+}^{d})$ , $1 < q \leq \infty$ . 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.

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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

Vol. 13, No. 4, 2020