Local existence for the free boundary problem for the non-relativistic and relativistic compressible Euler equations with a vacuum boundary condition

摘要

We study the free boundary problem for the equations of compressible Eulerequations with a vacuum boundary condition. Our main goal is to recover inEulerian coordinates the earlier well-posedness result obtained by Lindblad[Lindblad H., Commun. Math. Phys. 260 (2005), 319-392] for the isentropic Eulerequations and extend it to the case of full gas dynamics. For technicalsimplicity we consider the case of an unbounded domain whose boundary has theform of a graph and make short comments about the case of a bounded domain. Weprove the local-in-time existence in Sobolev spaces by the technique appliedearlier to weakly stable shock waves and characteristic discontinuities. Itcontains, in particular, the reduction to a fixed domain, using the "goodunknown" of Alinhac, and a suitable Nash-Moser-type iteration scheme. A certainmodification of such an approach is caused by the fact that the symbolassociated to the free surface is not elliptic. This approach is still directlyapplicable to the relativistic version of our problem in the setting of specialrelativity and we briefly discuss its extension to general relativity.