The Existence and Stability of Shock Fronts in Several Space Variables
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The phenomenon of shock wave formation described in the previous chapter indicates the necessity for studying discontinuous weak solutions of systems of conservation laws. Here we turn to the construction of discontinuous solutions of conservation laws in several space variables with an emphasis on the physical equations of compressible fluid flow. We concentrate on the rigorous short-time existence and structural stability of shock fronts in several space variables -- these are the simplest multi-D nonlinear progressing wave patterns, and in Section 4.1 we introduce the basic preliminary facts for this problem. We restrict our discussion for two reasons: 1) shock fronts are the most important nonlinear wave patterns in compressible fluid flow and other systems of conservation laws; 2) these special wave patterns are the only inherently discontinuous waves in multi-D for m × m systems where a rigorous theory has been developed (, ).
KeywordsWeak Solution Shock Front Mixed Problem Uniform Stability Linear Hyperbolic Equation
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