Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations

Abstract

The multidimensional piston problem is a special initial-boundary value problem. The boundary conditions are given in two conical surfaces: one is the boundary of the piston, and the other is the shock whose location is to be determined later. In this paper, we are concerned with spherically symmetric piston problem for the relativistic Euler equations. A local shock front solution with the state equation p = a 2 ρ, a is a constant and has been established by the Newton iteration. To overcome the difficulty caused by the free boundary, we introduce a coordinate transformation to fix it and employ the linear iteration scheme to establish a sequence of approximate solutions to the auxiliary problems by iteration. In each step, the value of the solution of the previous problem is taken as the data to determine the solution of the next problem. We obtain the existence of the original problem by establishing the convergence of these sequences. Meanwhile, we establish the convergence of the local solution as c → ∞ to the corresponding solution of the classical non-relativistic Euler equations.