The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas

We study the regularity of sonic curves to a two-dimensional Riemann problem for the nonlinear wave system of Chaplygin gas, which is an essential step for the global existence of solutions to the two-dimensional Riemann problems. As a result, we establish the global existence of uniformly smooth solutions in the semi-hyperbolic patches up to the sonic boundary, where the degeneracy of hyperbolicity occurs. Furthermore, we show the C^1 regularity of sonic curves.Comment: 15 pages, 3 figure

Supersonic flow of Chaplygin gas past a delta wing

We consider the problem of supersonic flow of a Chaplygin gas past a delta wing with a shock or rarefaction wave attached to the leading edges. The flow under study is described by the three-dimensional steady Euler system. In conical coordinates, this problem can be reformulated as a boundary value problem for a nonlinear equation of mixed type. The type of this equation depends fully on the solutions of the problem itself, and thus it cannot be determined in advance. We overcome the difficulty by establishing a crucial Lipschitz estimate, and finally prove the unique existence of the solution via the method of continuity.Comment: 32 pages, 10 figure