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Nonlinear oscillations of hyperbolic systems: Methods and qualitative results. (Oscillations non linéaires des systèmes hyperboliques: Méthodes et résultats qualitatifs.) (French) Zbl 0810.35060

Summary: The author studies the Cauchy problem for hyperbolic systems of first order conservation laws \(\partial_ t u+ \partial_ x f(u)=0\), with an oscillating sequence of initial data. The oscillations have an \(O(1)\) amplitude; one may think to \(u^ \varepsilon (x,0)= a(x; {x/ \varepsilon})\), where \(a(x,.)\) is periodic. It turns out that one part of the initial oscillations are killed because of genuine nonlinearity. But the most of the physically relevant systems are endowed with at least one linearly degenerate characteristic field, which allows oscillations of a part of the field \(u\).

MSC:

35L65 Hyperbolic conservation laws
76E30 Nonlinear effects in hydrodynamic stability
76Q05 Hydro- and aero-acoustics
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