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Stabilité des chocs faibles. (Stability of weak shocks). (French) Zbl 0702.35163
Sémin. Équations Dériv. Partielles 1988-1989, Exp. No. 20, 10 p. (1989).
The author is interested in the stability of weak shocks associated with symmetric hyperbolic systems of conservations laws, \(\partial u/\partial t+\sum_{v}A_ v(u)\partial u/\partial x_ v=0\). By a change of coordinates, the shock front is fixed at \(x_ n=0\); the corresponding conservation laws and jump conditions are thus modified and represent an initial boundary value problem in \(x_ n>0\) and \(x_ n<0\). This system then is linearized.
The notion of a uniformy stable shock, as characterized by Majda, requires solutions of this linearized problem to satisfy an a priori energy estimate which the author finds unsuitable for weak shocks. By introducing additional weak hypotheses on the structure of the problem, the author is able to extend this estimate in a manner suitable for discussion of the uniform stability of weak shocks.
Reviewer: John Crow
MSC:
35L67 Shocks and singularities for hyperbolic equations
35L60 First-order nonlinear hyperbolic equations
Full Text: Numdam EuDML