an:04115145
Zbl 0681.35057
Nakane, Shizuo
Formation of shocks for a single conservation law
EN
SIAM J. Math. Anal. 19, No. 6, 1391-1408 (1988).
00169382
1988
j
35L65 35L67 35C99
equation of scalar conservation law; singularities of; \(C^{\infty }\)- mappings; stable manifold; entropy condition; method of characteristics
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.
V.A.Yumaguzhin