Two-dimensional Euler equations in a time dependent domain
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.
uniqueness and regularity of weak solutions
incompressible fluid in a time dependent domain
scientific article; zbMATH DE number 1474623