Spreading of the free boundary of relativistic Euler equations in a vacuum

Thomas C. Sideris in [J. Differential Equations 257 (2014), no. 1, 1–14] showed that the diameter of a region occupied by an ideal fluid surrounded by vacuum will grow linearly in time provided the pressure is positive and there are no singularities. In this paper, we generalize this interesting result to isentropic relativistic Euler equations with pressure $p = \sigma^2 \rho$. We will show that the results obtained by Sideris still hold for relativistic fluids. Furthermore, a family of explicit spherically symmetric solutions is constructed to illustrate our result when $\sigma = 0$, which is different from Sideris’s self-similar solution.

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Received 26 October 2015

Accepted 24 August 2017

Published 25 March 2019