Semi-hyperbolic patches of solutions to the two-dimensional nonlinear wave system for Chaplygin gases

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

“…In fact, we do not know exactly how far we can extend the C 1 solution in Theorem 1.1 towards the sonic boundary. In this paper, we improve the results of [12] in the sense that the solutions in the semi-hyperbolic patches are uniformly smooth up to the sonic curve, where the degeneracy of hyperbolicity occurs, and that the sonic curve is C 1 continuous. In fact, the smoothness of sonic curve is a necessary requirement for the global existence of solutions in the region of OABD in Figure 2 for the future research.…”

“…Let W := (R, S, p), and we also denote the arcs AB and BC by Γ + and Γ − , respectively. For given boundary conditions on Γ ± , the global existence and a priori C 1 estimate of W in the domain ABC have been obtained in [12] as follows: [12]). Assume that two constant states (p 1 , m 1 , n 1 ) and (p 4 , m 4 , n 4 ) are given with p 1 < p 4 < 0 and a planar wave connects these two constant states in the domain ξ > 0.…”

Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations

“…For semi-hyperbolic patches of solutions to some other type of hyperbolic conservation laws, the readers can see [9,10,11]. Song et al [24,28] also studied the regularity of semi-hyperbolic patches near sonic lines.…”

The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations

“…We also refer the reader to [6,25,26,27] for the construction of classical sonic-supersonic solutions to the Euler system. The above works [7,6,21,16,19,24,25,26,27] on building smooth solutions are based on the idea of characteristic decomposition which is a powerful tool revealed in [4], see, e.g., [1,9,10,11,13,14,15,18] for more applications.…”