We study the 1D piston problem for the isentropic relativistic Euler equations when the total variations of the initial data and the speed of the piston are sufficiently small. Employing a modified Glimm scheme, we establish the global existence of shock front solutions including a strong shock without restriction on the strength. In particular, we give some uniform estimates on the perturbation waves, the reflections of the perturbation waves on the piston and the strong shock. Meanwhile, we consider the convergence of the entropy solutions as the light speed c → +∞ to the corresponding entropy solutions of the classical non-relativistic isentropic Euler equations.
Received 31 August 2012Accepted 31 January 2013Published online 14 March 2013
Min Ding would like to thank Professor Gui-Qiang Chen for his precious guidance and suggestions, also express her sincere gratitude to OxPDE Center, Mathematical Institute for its hospitality when she visits University of Oxford. Min Ding's research was supported in part by China Scholarship Council (No: 2009623053), and the EPSRC Science and Innovation Award to the Oxford Center for nonlinear PDE (No: EP/E035027/1). Yachun Li's research was supported in part by the National Natural Science Foundation of China under Grant Nos. 10971135 and 11231006.
Article outline: I. INTRODUCTION II. NONLINEAR ELEMENTARY WAVES III. THE BACKGROUND SOLUTION AND PISTON RIEMANN PROBLEM A. Background solution B. Piston Riemann problem IV. CONSTRUCTION OF THE APPROXIMATE SOLUTIONS V. ESTIMATES ON LOCAL INTERACTIONS A. Δ covers neither x = bΔ(t) nor x = sΔ(t) B. Δ covers part of x = bΔ(t) but none of x = sΔ(t) C. Δ covers part of both x = bΔ(t) and x = sΔ(t) D. Δ covers part of x = sΔ(t) but none of x = bΔ(t) (case I): Weak wavesinteract with the strong shock from the left E. Δ covers part of x = sΔ(t) but none of x = bΔ(t) (case II): Weak wavesinteract with the strong shock from the right VI. MONOTONICITY OF THE GLIMM FUNCTIONAL AND CONVERGENCE
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