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

Abstract

Zero viscosity limits are central to the study of classical shock waves. By identifying the correct physical (Lax admissible) shocks, they are a cornerstone in the design of analytical and numerical schemes. For relativistic fluid flow, however, the underlying dissipation mechanism, based on the Euclidean Laplace operator (so-called ``artificial viscosity''), violates Lorentz invariance, the fundamental principle of Special Relativity ensuring the speed of light bound. In this paper we show that replacing the Laplacian on conserved quantities by the wave operator on the fluid four-velocity alone, (not involving the density), 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, and we establish their dissipativity by proving decay of Fourier Laplace modes near steady states. 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 an efficient 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.