Search

We consider solutions of the two-dimensional compressible (isentropic) Euler equations that are steady and self-similar. They arise naturally at interaction points in genuinely multidimensional flow. We characterize the possible solutions in the class of flows $L^\infty$-close to a constant supersonic background. As a special case we prove that solutions of one-dimensional Riemann problems are unique in the class of small $L^\infty$ functions. We also show that solutions of the backward-in-time Riemann problem are necessarily BV.

We consider a particular instance of reflection of shock waves in self-similar compressible flow. We prove that local self-similar regular reflection (RR) cannot always be extended into a global flow. Therefore the detachment criterion is not universally correct. More precisely, consider the following "angle condition": the tangent of the strong-type reflected shock meets the opposite wall at a sharp or right downstream side angle. In cases where the condition is violated and the weak-type reflected shock is transonic, we show that global RR does not exist. Combined with earlier work we have shown that none of the classical criteria for RR-MR transition is universally correct. A new criterion is proposed. Moreover, we have shown that strong-type RR is unstable, in the sense that global RR cannot persist under perturbations to one side. This yields a definite answer to the weak-strong problem because earlier work shows stability of weak RR in the same sense., Comment: Formatting corrected