Two-dimensional Euler equations in a time dependent domain
type: Item
instance of: scholarly article
title: Two-dimensional Euler equations in a time dependent domain
zbMATH Open document ID: 0958.35096
author: Cheng He
author: Ling Hsiao
published in: Journal of Differential Equations
publication date: 2001-04-02T00:00:00Z
review text: The authors consider the problem of existence, uniqueness and regularity of weak solutions of the two-dimensional Euler equations for an ideal incompressible fluid in a time-dependent domain. First of all, it is proved that if the time-dependent domain is diffeomorphic to a cylindrical one, if the initial velocity is square integrable and the initial vorticity is \(r\)-integrable for some \(r\) greater than 1 and if the external force vector field is sufficiently regular, then there exists at least one weak solution of the two-dimensional Euler equations. Moreover, if the external force and the movement of the domain are periodic with respect to time, then there exists a weak periodic solution of these equations. Next, the case when the domain moves along the path lines issued from the initial domain is considered. The existence of weak solution is proved again, this time for bounded initial vorticity. The uniqueness of these solutions is obtained only in the absence of external forces. The paper ends with some standard results on the regularity of the weak solutions.
reviewed by: Mihai Pascu
Mathematics Subject Classification ID: 35Q05
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35M10
zbMATH DE Number: 1474623
zbMATH Keywords: existence
zbMATH Keywords: uniqueness and regularity of weak solutions
zbMATH Keywords: two-dimensional Euler equations
zbMATH Keywords: incompressible fluid in a time dependent domain
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3702
OpenAlex ID: W2028276015
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DOI: 10.1006/JDEQ.1999.3702
description: scientific article; zbMATH DE number 1474623