A symmetrization of the relativistic Euler equations with severalspatial variables

Philippe G. LeFloch; Seiji Ukai

doi:10.3934/krm.2009.2.275

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2009, Volume 2, Issue 2: 275-292. Doi: 10.3934/krm.2009.2.275

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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
2.
17-26 Iwasaki, Hodogaya, Yokohama 240-0015

Received: November 30, 2008

Revised: December 31, 2008

Published: May 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 35L65; Secondary: 76N10.

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