Systems of hyperbolic conservation laws with a resonant moving source

In this paper, we study the stability of a single transonic shock wave solution to the hyperbolic conservation laws with a resonant moving source. Compared with the previous results [W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075–1098; T.P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys. 83 (2) (1982) 243–260] on this stability problem, in this paper, the transonic ith shock is assumed to be relatively strong and stable in the sense of Majda. Then the framework of [M. Lewicka, L1 stability of patterns of non-interacting large shock waves, Indiana Univ. Math. J. 49 (4) (2000) 1515–1537; M. Lewicka, Stability conditions for patterns of noninteracting large shock waves, SIAM J. Math. Anal. 32 (5) (2001) 1094–1116 (electronic)] can be applied. A new criterion is obtained to test whether such a shock is time asymptotically stable or not. And by constructing the Liu–Yang functional, one can prove the L1 stability of the shock under the stability condition. This is an extension of the result [S.-Y. Ha, T. Yang, L1 stability for systems of hyperbolic conservation laws with a resonant moving source, SIAM J. Math. Anal. 34 (5) (2003) 1226–1251 (electronic); W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075–1098] to a more general case.