Global solutions for the ultra-relativistic Euler equations

Abstract

A front tracking scheme for the ultra-relativistic Euler equations is introduced. This scheme is based on piecewise constant approximations to the front tracking Riemann solutions, where continuous rarefaction waves are approximated by finite collections of discontinuities, so-called non-entropy shocks. We study the interaction estimates of the generalized shocks (entropy and nonentropy shocks) of the ultra-relativistic Euler equations and the outcoming asymptotic Riemann solution. Moreover we use a new function to measure the strengths of the waves of the ultra-relativistic Euler equations. This function has the important implication that the strength is non increasing for the interactions of the generalized shocks. This enables us to define a new kind of total variation of a solution. The main application of this scheme, is proving the global existence of weak solutions for the ultra relativistic Euler equations.