Two-dimensional incompressible ideal flows in a noncylindrical material domain

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The purpose of this work is to prove existence of a weak solution of the two dimensional incompressible Euler equations on a noncylindrical domain consisting of a smooth, bounded, connected and simply connected domain undergoing a prescribed motion. We prove existence of a weak solution for initial vorticity in Lᵖ, for p>1. This work complements a similar result by C. He and L. Hsiao, who proved existence assuming that the flow velocity is tangent to the moving boundary, see [JDE v. 163 (2000) 265--291].