Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires. (Asymptotic expansion of exact solutions of quasilinear hyperbolic systems). (French) Zbl 0780.35017

Guès, Olivier

Asymptotic Anal. 6, No. 3, 241-269 (1993).

The existence of rapidly oscillating stable solutions for multidimensional symmetrizable quasilinear hyperbolic systems is proved. This is used to provide a rigorous justification of asymptotic expansions due to Choquet-Bruhat. The developments are of the form \(u_ \varepsilon(x)\sim u_ 0(x)+ \sum^ \infty_{j=1} \varepsilon^ j u^ j\bigl( x, {{\varphi(x)} \over \varepsilon} \bigr)\), where \(\varepsilon>0\) is a small parameter and \(u^ j\) is a \(2\pi\)-periodic function in the second variable. Approximate solutions of some Cauchy problem are obtained and then used in order to get exact solutions. One of the main ingredients of the proofs are a priori estimates for a linearized problem in weighted Sobolev spaces.

Reviewer: J.Hernandez (Madrid)

MSC:

35C20	Asymptotic expansions of solutions to PDEs
35B25	Singular perturbations in context of PDEs
35L60	First-order nonlinear hyperbolic equations
35L40	First-order hyperbolic systems

Keywords:

oscillating solutions; approximate solutions; quasilinear hyperbolic systems; asymptotic expansions; Cauchy problem; a priori estimates; weighted Sobolev spaces