Abstract
AbstractWe analyze the relativistic Euler equations of conservation laws of baryon number and momentum with a general pressure law. The existence of global-in-time bounded entropy solutions for the system is established by developing a compensated compactness framework. The proof relies on a careful analysis of the entropy and entropy-flux functions, which are represented by the fundamental solutions of the entropy and entropy-flux equations for the relativistic Euler equations. Based on a careful entropy analysis, we establish the compactness framework for sequences of both exact solutions and approximate solutions of the relativistic Euler equations. Then we construct approximate solutions via the vanishing viscosity method and employ our compactness framework to deduce the global-in-time existence of entropy solutions. The compactness of the solution operator is also established. Finally, we apply our techniques to establish the convergence of the Newtonian limit from the entropy solutions of the relativistic Euler equations to the classical Euler equations.
Funder
Engineering and Physical Sciences Research Council
Royal Society
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Geometry and Topology,General Physics and Astronomy,Mathematical Physics,Analysis
Reference27 articles.
1. Ball, J. M.: A version of the fundamental theorem for Young measures, In: PDEs and Continuum Models of Phase Transitions (Nice, 1988), 207–215, Lecture Notes in Physics, 344, Springer: Berlin (1989)
2. Chen, G.-Q.: Convergence of the Lax–Friedrichs scheme for isentropic gas dynamics (III). Acta Math. Sci. 6, 75–120 (1986) (in English); 8, 243–276 (1988) (in Chinese)
3. Chen, G.-Q.: The compensated compactness method and the system of isentropic gas dynamics, Preprint MSRI-00527-91. Math. Sci. Res. Inst., Berkeley (1990)
4. Chen, G.-Q., LeFloch, P.G.: Compressible Euler equations with general pressure law. Arch. Ration. Mech. Anal. 153, 221–259 (2000)
5. Chen, G.-Q., LeFloch, P.G.: Existence theory for the isentropic Euler equations. Arch. Ration. Mech. Anal. 166, 81–98 (2003)