Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires

Abstract

We prove existence and stability of high-frequency oscillating solutions for multidimensional quasilinear hyperbolic systems, justifying the asymptotic developments of Choquet-Bruhat (1969). We are concerned with solutions depending on a (small) parameter ε, which admit a development of the form u_ε(x)=u₀(x)+εu¹(x,ϕ(x)/ε)+…+ε^M−1u^M−1(x,ϕ(x)/ε)+O(ε^M),(ε→0), where u₀ is a given solution of the system, profiles u^j(x,θ) are (smooth) 2π-periodic with respect to the fast variable θ∈R, and the integer M is of the order of n/2, n being time-space dimension. We also show that, an arbitrary T>0 being given, one can built such oscillating solutions that remain regular on the interval of time [0,T] (for every ε>0 enough small). These results are obtained by mean of more general approximation theorems, adapted to the justification of such asymptotic developments.