Stability of Riemann solutions with large oscillation for the relativistic Euler equations

Abstract

We are concerned with entropy solutions of the 2 Â 2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L N -BV loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily large L 1 -L N -BV loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L N and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L N with arbitrarily large oscillation. r 2004 Elsevier Inc. All rights reserved. MSC: primary: 35B40; 35A05; 76Y05; secondary: 35B35; 35L65; 85A05

Global entropy solutions to the relativistic Euler equations for a class of large initial data

“…System (1.1) models the dynamics of plane waves in special relativistic fluids (cf. e.g [5,16,17,18,19]) in a two dimensional Minkowski space-time (x 0 , x 1 ): div T = 0, with the stress-energy tensor for the fluid:…”

“…e.g. [5,7,15]). System (1.1) fits into the following general form of hyperbolic system of conservation laws ∂ t U + ∂ x F (U ) = 0, (1.8) by setting…”

Dependence of entropy solutions in the large for the Euler equations on nonlinear flux functions

“…Due to the Lorenz invariance, if there are two states (ρ L , u L ) and (ρ R , u R ) connected by a shock, we can assume the velocity state of the left-hand side of the shock in the barred coordinates isū L = 0. Hence the Lax entropy and Rankine-Hugoniot conditions imply (see [6,8,17,19]…”

Global radial solutions to 3D relativistic Euler equations for non-isentropic Chaplygin gases

“…Recently, Ding et al [12] considered the full Euler equations (1.11) in two space dimensions, they proved the global existence of smooth spherically symmetric solutions by applying Alinhac's ghost weight energy and the idea of Godin [14], in which he proved the smooth spherically symmetric solutions for 3D full compressible Euler equations (1.11) of Chaplygin gases. For relativistic Euler equations (1.10) of Chaplygin gases, known results are mainly on the Riemann problems, we refer to [7,8,16]. Due to the complexity of relativistic system (1.8) or (1.10), there are no global existence results for more than one space dimensions.…”