Long time existence of a class of perturbations of planar shock fronts for second order hyperbolic conservation laws. (English) Zbl 0719.35056

By reducing the shock front problem of the form \[ (1)\quad \sum_{0\leq i\leq N}\partial_ i(H^ i(\phi '))=0,\text{ where } \phi '=(\partial_ 0\phi,\partial_ x\phi),\quad \partial_ i=\partial /\partial x_ i,\quad \partial_ x\phi =(\partial_ 1\phi,...,\partial_ N\phi), \] to a mixed problem in a quarter space via the partial hodograf transform a global planar shock front solution satisfying Lax type entropy conditions is studied.
It is shown that small \(C^{\infty}\) compatible perturbations with compact support of the data give rise to shock fronts with long lifespan (if \(N\geq 5)\) which remain stable and almost planar.
Reviewer: K.Zlateva (Russe)


35L65 Hyperbolic conservation laws
35B20 Perturbations in context of PDEs
35L67 Shocks and singularities for hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
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