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June  2009, 2(2): 275-292. doi: 10.3934/krm.2009.2.275

A symmetrization of the relativistic Euler equations with several spatial variables

Philippe G. LeFloch 1, and  Seiji Ukai 2,

1. 

Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252 Paris, France
2. 

17-26 Iwasaki, Hodogaya, Yokohama 240-0015, Japan

Received  December 2008 Revised  January 2009 Published  May 2009

We consider the Euler equations governing relativistic compressible fluids evolving in the Minkowski spacetime with several spatial variables. We propose a new symmetrization which makes sense for solutions containing vacuum states and, for instance, applies to the case of compactly supported solutions which are important to model star dynamics. Then, relying on these symmetrization and assuming that the velocity does not exceed some threshold and remains bounded away from the light speed, we deduce a local-in-time existence result for solutions containing vacuum states. We also observe that the support of compactly supported solutions does not expand as time evolves.
Keywords: symmetric hyperbolic system, vacuum, relativistic fluid, local well-posedness theory., Euler equations.
Mathematics Subject Classification: Primary: 35L65; Secondary: 76N1.
Citation: Philippe G. LeFloch, Seiji Ukai. A symmetrization of the relativistic Euler equations with several spatial variables. Kinetic & Related Models, 2009, 2 (2) : 275-292. doi: 10.3934/krm.2009.2.275
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