Well and ill-posedness of free boundary problems to relativistic Euler equations

Abstract

In this paper, via the regularity of sonic speed, we are concerned with the well and ill-posedness problems of the relativistic Euler equations with free boundary. First, we deduce the physical vacuum condition of relativistic Euler equations, which means that the sonic speed [Formula: see text] behaves like a half power of distance to the vacuum boundary [Formula: see text], satisfying [Formula: see text], it belongs to H[Formula: see text]lder continuous. Then, for [Formula: see text], this case means that the sonic speed belongs to [Formula: see text] smooth across the vacuum boundary, it is proved from both Lagrangian and Eulerian coordinates points of view. Finally, for the cases [Formula: see text] and [Formula: see text], the boundary behaviors are verified ill-posed by the unbounded acceleration of the fluid near the vacuum boundary. In this paper, the uniform bounds of velocity [Formula: see text] with respect to [Formula: see text] and the upper bounds for the square of sonic speed [Formula: see text] are very important in the proof of no matter whether well or ill-posedness because this will enable us to avoid many difficulties in the mathematical structure of relativistic fluids especially near the vacuum boundary. It is our innovation that distinguishes from non-relativistic Euler equations [J. Jang and N. Masmoudi, Well and ill-posedness for compressible Euler equations with vacuum, J. Math. Phys. 53 (2012) 1–11].