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Propagation des oscillations dans deux classes de systèmes hyperboliques (2\(\times 2\) et 3\(\times 3)\). (Propagation of oscillations in two classes of hyperbolic systems). (French) Zbl 0649.35058

We study in this note the solutions of two types of hyperbolic systems of conservation laws with oscillating data.
The first one (a \(2\times 2\) system) has only one linearly degenerate eigenvalue. Using the results of R. J. di Perna related to genuinely nonlinear fields, one can describe the propagation of oscillations (which appear in only one direction) with an integro-differential system for which one of the two unknowns is a field depending of \(y\in]0,1[\), in addition of x and t.
The second system is a linearly degenerate \(3\times 3\) system. We apply the theory of compensated compactness, due to L. Tartar and F. Murat and, in the same way as above, we show that the inital oscillations can propagate; this propagation is then described with a relaxed system of 3 unknowns.
Reviewer: M.Bonnefille

MSC:

35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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