Existence of multidimensional shock waves and phase boundaries

In this paper, a kind of Riemann problem for the Euler equations in a van der Waals fluid is considered. We constructed the weak solution in multidimensional space which contains one shock front and one subsonic phase boundary. We mainly follow the arguments of Majdaʹs [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1–95; A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1–93] and Métivierʹs [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux dʹespace, Trans. Amer. Math. Soc. 296 (1986) 431–479] work. The linear stability results are based on Majdaʹs [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1–95] work for the single shock front and Wang and Xinʹs [Y.-G. Wang, Z. Xin, Stability and existence of multidimensional subsonic phase transitions, Acta Math. Appl. Sin. 19 (2003) 529–558] work for the single phase boundary. The initial boundary value problem concerned in this paper is different from the boundary value problem for double shock fronts concerned in [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux dʹespace, Trans. Amer. Math. Soc. 296 (1986) 431–479], we slightly modified Métivierʹs frame work to establish the existence for the solution to the nonlinear problem.