Shock waves with irrotational Rankine-Hugoniot conditions (Q6602246)

This paper examines the linear stability of shock waves in the Euler system of gas dynamics under a new set of boundary conditions. The author demonstrate that the shock wave is linearly stable if and only if the usual Lax shock condition is satisfied. This result aligns with the well-established conclusion for shock front solutions in the Euler system with the traditional Rankine-Hugoniot free boundary conditions.\N\NBuilding on this linear stability result, it becomes possible to establish the existence of shock wave solutions for the Euler system and the second-order nonlinear wave equation. Notably, the double shock wave solutions for the nonlinear wave equation can be derived using a simpler method compared to the approach in [\textit{M. Mnif} and \textit{I. P. E. I. Sfax}, Commun. Partial Differ. Equations 22, No. 9--10, 1589--1627 (1997; Zbl 0904.35050)]. This work also paves the way for studying solutions containing a full set of shock waves or rarefaction waves, thereby enabling a complete resolution of generalized Riemann problems for such nonlinear wave equations. However, in this paper, the author focus solely on the discussion of linear stability. The existence of solutions is briefly outlined in Section 4 and will be addressed in detail in a future publication.\N\NFor simplicity, the discussion is restricted to the two-dimensional case, though the three-dimensional model can be formulated similarly. The stability analysis in three dimensions can follow the same approach as outlined in Section 3. While the computations are conceptually straightforward, they are technically cumbersome. These results will be presented in a future work focused on the three-dimensional non-isentropic Euler system, along with a comparative study of shocks under traditional Rankine-Hugoniot conditions.