Multi-dimensional shock fronts for second order wave equations. (English) Zbl 0632.35047

The authors study the short-time existence and structural stability of multidimensional shock fronts satisfying scalar second order quasi-linear wave equations. Their long rigorous proof given here for this special class of equations is simpler than those for general first order systems presented previously by the first author [Mem. Am. Math. Soc. 275, 95 p. (1983; Zbl 0506.76075) and 281, 92 p. (1982; Zbl 0517.76068)], although the basic ideas of these papers are here utilized.
The elegant strategy of their proof consists basically in the reduction of this shock front problem to an existence problem for a single scalar function defined on one side of the shock front and then in the introduction of a suitable partial hodograph transformation which leads to a nonlinear mixed problem for a single unknown function. This transformation (which has been used earlier by different authors in the study of elliptic free boundary value problems) is not only a remarkable simplification of the problem under consideration, but seems also to be appropriate tool for solving second order equations for which the previous techniques for the first order systems given in Majda’s papers quoted above are not successful. Then a fixed point iteration scheme is used. A special emphasis is given to the multidimensional shock fronts for the equations of isontropic irrotational compressible fluid which represents an important physical example for their theory.
Reviewer: P.Pucci


35L70 Second-order nonlinear hyperbolic equations
35L67 Shocks and singularities for hyperbolic equations
76G25 General aerodynamics and subsonic flows
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B35 Stability in context of PDEs
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