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

##### 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