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2020 Estimates for the Navier–Stokes equations in the half-space for nonlocalized data
Yasunori Maekawa, Hideyuki Miura, Christophe Prange
Anal. PDE 13(4): 945-1010 (2020). 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 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.

Citation

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Yasunori Maekawa. Hideyuki Miura. Christophe Prange. "Estimates for the Navier–Stokes equations in the half-space for nonlocalized data." Anal. PDE 13 (4) 945 - 1010, 2020. https://doi.org/10.2140/apde.2020.13.945

Information

Received: 16 November 2017; Revised: 4 March 2019; Accepted: 18 April 2019; Published: 2020
First available in Project Euclid: 25 June 2020

zbMATH: 07221194
MathSciNet: MR4109897
Digital Object Identifier: 10.2140/apde.2020.13.945

Subjects:
Primary: 35A01, 35A02, 35B44, 35B53, 35Q30
Secondary: 35C99, 76D03, 76D05

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.13 • No. 4 • 2020
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