Vol. 13, No. 5, 2020

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Well-posedness of the hydrostatic Navier–Stokes equations

David Gérard-Varet, Nader Masmoudi and Vlad Vicol

Vol. 13 (2020), No. 5, 1417–1455
Abstract

We address the local well-posedness of the hydrostatic Navier–Stokes equations. These equations, sometimes called reduced Navier–Stokes/Prandtl, appear as a formal limit of the Navier–Stokes system in thin domains, under certain constraints on the aspect ratio and the Reynolds number. It is known that without any structural assumption on the initial data, real-analyticity is both necessary and sufficient for the local well-posedness of the system. In this paper we prove that for convex initial data, local well-posedness holds under simple Gevrey regularity.

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Keywords
fluid mechanics, Navier–Stokes equations
Mathematical Subject Classification 2010
Primary: 35Q30
Secondary: 35Q35
Milestones
Received: 12 April 2018
Revised: 2 May 2019
Accepted: 11 June 2019
Published: 27 July 2020
Authors
David Gérard-Varet
Institut de Mathématiques de Jussieu-Paris Rive Gauche
Université Paris Diderot
Institut Universitaire de France
Paris
France
Nader Masmoudi
Courant Institute of Mathematical Sciences
New York University
New York
United States
Vlad Vicol
Department of Mathematics
Princeton University
Princeton, NJ
United States