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Formation of shocks for a single conservation law. (English) Zbl 0681.35057
The initial value problem for the equation of scalar conservation law is considered \[ \partial u/\partial t+\sum^{n}_{i=1}\partial f_ i(u)/\partial x_ i=0\quad in\quad \{(t,x)\in {\mathbb{R}}^{n+1},\quad t>0\},\quad u(0,x)=\phi (x)\quad on\quad {\mathbb{R}}^ n, \] where \(f=(f_ 1,...,f_ n)\) is a \(C^{\infty}\)-mapping \({\mathbb{R}}\to {\mathbb{R}}^ n,\) \(\phi\) is a real-valued \(C^{\infty}\) rapidly decreasing function on \({\mathbb{R}}^ n\). The solution of this problem is concretely constructed by the method of characteristics. Its structure as a multivalued function is completely revealed by virtue of the theory of singularities of \(C^{\infty}\)-mappings. Shocks appear in this process. Shock surfaces are constructed by using the stable manifold theory.
Reviewer: V.A.Yumaguzhin

MSC:
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
35C99 Representations of solutions to partial differential equations
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