The expansion of a non‐ideal gas around a sharp corner for 2‐D compressible Euler system

Abstract

In this paper, we consider the problem of expansion of a non‐ideal gas turning around corresponding large or small sharp corner for 2‐D compressible Euler system. We focus on extending the results of this problem for polytropic gas to that for a more realistic gas, that is, van der Waals gas. In this case, rarefaction waves, shock wave, fan–jump composite waves, or fan–jump–fan composite waves may be appeared. The flow expands to vacuum state when the inclination angle is small enough, that is, a zone of cavitation will come into being. Otherwise, the flow will arrive at a zone of constant state. The corresponding problem can be transformed into interaction of a planar rarefaction wave with a planar centered wave and interaction of a simple wave with rigid wall, which are actually characteristic boundary value problems for 2‐D self‐similar Euler system. The estimates of solution are obtained by making use of characteristic analysis, corresponding characteristic decompositions and invariant region of solution. Furthermore, by extending the local solution, and combining with those estimates and hyperbolicity, the existence of global classical solution up to the interface of non‐ideal gas with vacuum is obtained.