Shock solutions of a 3D spherical piston problem for the compressible Euler equations

Abstract

This paper establishes the global well-posedness of shock solutions for a 3D spherical piston problem to the compressible Euler equations. The main mathematical challenges are the nonlinear coupling between the free shock boundary and the moving piston, alongside potential gradient catastrophes induced by spherical geometric source terms. To overcome these difficulties, we reformulate the problem using Riemann invariants and introduce a tailored directional derivative along the unperturbed piston trajectory. Then we can extend the local solution to construct a global piecewise smooth solution including a single shock under some moderate assumptions.