Nonrelativistic limits for the 1D relativistic Euler equations with physical vacuum

Abstract

We prove the nonrelativistic limits of the local smooth solutions to the free boundary value problem of the 1D relativistic Euler equations, when the mass energy density includes the vacuum states at the free boundary. We successfully overcome the strong nonlinearity and the degenerate difficulty of the relativistic Euler equations caused by the Lorentz factor and the vacuum occurring on the moving boundary, respectively. Moreover, the smooth solutions of the relativistic Euler equations converge to the solutions of the classical compressible Euler equation, at the rate of \(\frac{1}{c^2}\).