A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary |
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Authors: | Mahir Hadžić Jared Speck |
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Institution: | 1. Department of Mathematics, King’s College London, London, UK;2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA |
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Abstract: | We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known as co-moving) coordinates, of local-in-time a priori estimates for the solution. The solution features a fluid-vacuum boundary, transported by the fluid four-velocity, along which the hyperbolicity of the equations degenerates. In this context, the relativistic Euler equations are equivalent to a degenerate quasilinear hyperbolic wave-map-like system that cannot be treated using standard energy methods. |
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