On local well-posedness of problems with characteristic free boundary for hyperbolic systems of conservation laws

Yu. L. Trakhinin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 79 (2024) no. 2, pp. 325-360

Résumé

The proof of the local-in-time existence and uniqueness of a smooth solution to a free boundary problem for a hyperbolic system of conservation laws has some additional difficulties if the free boundary is a characteristic of this system. They are connected with the loss of control of the normal derivatives and the possible non-ellipticity of the symbol of the free boundary. Another peculiarity of problems with characteristic free boundary is that usually a loss of derivatives of the coefficients and source terms occurs in a priori estimates for the corresponding linearized problems. Moreover, the boundary conditions in the linearized problem can be non-dissipative, which makes it difficult to use the energy method. We describe methods for overcoming these difficulties. Our main examples are free boundary problems for Euler's equations and the equations of ideal compressible magnetohydrodynamics, for which we review the results on their local well-posedness. Bibliography: 61 titles.

Détail

Keywords: hyperbolic system of conservation laws, free boundary problem, characteristic boundary, local existence and uniqueness theorem, loss of derivatives in a priori estimates, non-ellipticity of the boundary symbol, secondary symmetrization, tame estimates, Nash–Moser method.