Entropy solutions of the compressible Euler equations

Abstract

We consider the three-dimensional Euler equations of gas dynamics on a bounded periodic domain and a bounded time interval. We prove that Lax-Friedrichs scheme can be used to produce a sequence of solutions with ever finer resolution for any appropriately bounded (but not necessarily small) initial data. Furthermore, we show that if the density remains strictly positive in the sequence of solutions at hand, a subsequence converges to an entropy solution. We provide numerical evidence for these results by computing a sensitive Kelvin-Helmholtz problem.