Dissipative Symmetrizers of Hyperbolic Problems and Their Applications to Shock Waves and Characteristic Discontinuities

By introducing the notations of dissipative and strictly dissipative p-symmetrizers of initial-boundary-value problems for linear hyperbolic systems we formalize the dissipative integrals technique [A. Blokhin, Yu. Trakhinin, in Handbook of Mathematical Fluid Dynamics, Vol. 1, North-Holland, Amsterdam, 2002, pp. 545–652] applied earlier to shock waves and characteristic discontinuities for various concrete systems of conservation laws. This enables us to prove the local in time existence of shock-front solutions of an abstract symmetric system of hyperbolic conservation laws, provided that the corresponding constant coefficients linearized problem has a strictly dissipative p-symmetrizer. Our result does not, in particular, require the fulfillment of Majda’s block structure condition. A p-symmetrizer is, in some sense, a “secondary” Friedrichs symmetrizer for the symmetric system for the vector of p-derivatives of unknown functions, and the structure of p-symmetrizer takes into account (if applicable) the set of divergent constraints for the original system. After applying a p-symmetrizer, which is in general a set of matrices and vectors, the boundary conditions for a resulting symmetric system are dissipative (or strictly dissipative). We give concrete examples of p-symmetrizers. Our main examples are strictly dissipative 2-symmetrizers for shock waves in gas dynamics and magnetohydrodynamics. A general procedure for constructing a p-symmetrizer does not however exist. But, if it was somehow constructed, then we do not need to test the Lopatinski condition that is often connected with insuperable technical difficulties. As an illustration, we refer to the author’s recent result [Yu. Trakhinin, Arch. Ration. Mech. Anal., 177 (2005), pp. 331–366] for compressible current-vortex sheets for which the construction of a dissipative 0-symmetrizer has first enabled the finding of sufficient conditions for their weak linearized stability.