On the Motion of the Free Surface of a Compressible Liquid

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Johns Hopkins University
We prove a priori estimates for the compressible Euler equations modeling the motion of a liquid with moving physical vacuum boundary. The liquid is not assumed to be irrotational. But the physical sign condition needs to be assumed on the free boundary. The a priori energy estimates are in fact uniform in the sound speed $\kk$. As a consequence, we obtain the convergence of solution of the compressible Euler equations with a free boundary to solution of the incompressible equations, generalizing the result of Ebin \cite{Eb} to when you have a free boundary. In the incompressible case our energies reduces to those in \cite{CL} and our proof in particular gives a simplified proof of the estimates in Christodoulou-Lindblad \cite{CL} with improved error estimates. Since for an incompressible irrotational liquid with free surface there are small data global existence results, our result leaves open the possibility of long time existence also for slightly compressible liquids with a free surface. This thesis is based on the author's paper \cite{LL} (joint with H. Lindblad, accepted and to be published in C.P.A.M) and \cite{Lu}.
partial differential equations, free boundary problem