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Estimates for the
Navier–Stokes equations in the half-space for nonlocalized
data
Yasunori Maekawa, Hideyuki Miura and Christophe Prange |
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Vol. 13 (2020), No. 4, 945–1010
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Abstract |
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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 . We prove the analyticity of the Stokes semigroup in for . This follows from the analysis of the Stokes resolvent problem for data in , . 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
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Mathematical Subject Classification 2010
Primary: 35A01, 35A02, 35B44, 35B53, 35Q30
Secondary: 35C99, 76D03, 76D05
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Milestones
Received: 16 November 2017
Revised: 4 March 2019
Accepted: 18 April 2019
Published: 13 June 2020
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Authors |
| Yasunori Maekawa | |
| Department of Mathematics Kyoto University Kyoto Japan |
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| Hideyuki Miura | |
| Department of Mathematical and
Computing Sciences Tokyo Institute of Technology Tokyo Japan |
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| Christophe Prange | |
| CNRS, UMR 5251, IMB Université de Bordeaux Bordeaux France |
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