### Spatially Periodic Solutions in Relativistic Isentropic Gas Dynamics

Hermano Frid and Mikhail Perepelitsa

Abstract: We consider the initial value problem, with periodic initial data, for the Euler equations in relativistic isentropic gas dynamics, for ideal polytropic gases which obey a constitutive equation, relating pressure $p$ and density $\rho$, $p=\k^2\rho^\g$, with $\g\ge1$, $0<\k<c$, where $c$ is the speed of light. Global existence of periodic entropy solutions for initial data sufficiently close to a constant state follows from a celebrated result of Glimm and Lax (1970). We prove that given any periodic initial data of locally bounded total variation, satisfying the physical restrictions $0<\inf_{x\in\R}\rho_0(x)<\sup_{x\in\R}\rho_0(x)<+\infty$, $\|v_0\|_\infty<c$, where $v$ is the gas velocity, there exists a globally defined spatially periodic entropy solution for the Cauchy problem, if $1\le \g<\g_0$, for some $\g_0>1$, depending on the initial bounds. The solution decays in $L_{loc}^1$ to its mean value as $t\to \infty$.

Paper:
Available as PDF (392 Kbytes).
Author(s):
Hermano Frid, <hermano@impa.br>
Mikhail Perepelitsa, <mikhailp@math.northwestern.edu>
Publishing information: