Dissipative relativistic fluid flow: A simple Lorentz invariant causal model capturing entropy shocks in its zero viscosity limit

Abstract

Zero viscosity limits, by identifying the physical (Lax admissible) shocks, are a cornerstone in the design of analytical and numerical schemes for the study of classical shock waves. For relativistic fluid flow, however, the underlying Laplacian based dissipation mechanism (so-called ``artificial viscosity'') violates Lorentz invariance, the fundamental principle of Special Relativity. In this paper we show that replacing the Laplacian on conserved quantities by the wave operator on the fluid four-velocity alone, provides a simplest Lorentz invariant description of dissipative relativistic fluid flow. We prove the resulting equations are causal and well-posed in one spatial dimension. We establish non-linear dissipativity and global-in-time existence of smooth solutions with small data in $H^s\cap L^1$, based on a Kawashima compensated energy method; (this appears to be the first such result for relativistic dissipation in 1-D). Moreover, we prove shock waves have profiles (a unique viscous travelling wave approximation in $L^2$) if and only if the shock wave is Lax admissible, and we prove that entropy production of travelling wave solutions is positive if and only if they obey the speed of light bound. This establishes the dissipative relativistic Euler equations introduced in this paper as a viable model for the study of relativistic shock waves in the zero viscosity limit, both in analytical and numerical approaches, consistent with the laws of Relativity.