Well-Posedness of the Free Boundary Problem for the Compressible Euler Equations and the Incompressible Limit

Source abstract

In this paper, we study the free boundary problem of the compressible Euler equations in the Eulerian coordinates. By deriving the evolution equation of the free surface, we relate the Taylor stability condition to the hyperbolicity of this evolution equation. Our approach not only yields exact information of the free surface, but also gives a simple proof of the local well-posedness of the free boundary problem. This approach provides a unified framework to treat both compressible and incompressible free boundary problems. As a byproduct, we can also prove the incompressible limit.