A Supplement to Entropy Condition

The purpose of this article is to study the uniqueness and stability of steady flow with a shock front. When a uniform supersonic flow past a plane wedge with the vertex angle less than a critical value, there will be a plain shock front attached at the edge of the wedge. The location of the shock can be determined by the intersection of the ray with a given angle and the shock polar determined by the parameters of the coming flow. However, in most cases there will be two such intersections: one corresponds to the weaker oblique shock, the other corresponds to the stronger oblique shock. Both two satisfy the Rankine-Hugoniot condition and the entropy condition. Hence a natural question arises, which one is actually occurs? It is generally understood that the weaker one is stable and the stronger one is not, so that only the weak one could occur. However, there is not a convincing proof in mathematics so far. Since in the shock theory many problems are based on the study of this fundamental problem, for instance, the study of supersonic flow past a given body, the reflection or the interaction of shock waves etc., then, it is important and desirable to find a criterion as a supplement of the entropy condition to single out a physically reasonable shock front.