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Ondes de raréfaction pour des systèmes quasilinéaires hyperboliques multidimensionnels. (Rarefaction waves for multidimensional quasilinear hyperbolic systems). (French) Zbl 0669.35070
Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1988, Exp. No. 8, 7 p. (1988).
The work studies the existence of waves of rarefaction for the multidimensional quasilinear hyperbolic system of the type \[ (1)\quad \partial_ tv+A_ 1(v)\partial_ xv+A_ 2(v)\partial_ yv=0, \] where \(t\in {\mathbb{R}}\), \(x\in {\mathbb{R}}\) and \(A_ 2(v)\partial_ y=\sum A_ 2^{(j)}\partial_{y_ j}\) with \(y\in {\mathbb{R}}^{n-2}\). The initial data are supposed to be smooth on the complement of a smooth hypersurface \(\Gamma\) lying on the plane \(t=0\). The main characteristic property of the waves of rarefaction is that their initial data satisfy suitable compactibility condition on \(\Gamma\). The author introduces the important notion of the index d of compatible initial data. The index is closely related to the loss of regularity of the solution to (1). The system (1) is supposed to be symmetrisable in the sense that \(SA_ 1\) and \(SA_ 2\) are symmetric matrices for some positively defined matrix S. The main result of the work guarantees the existence and uniqueness of the wave of rarefaction. The proof is based on the application of the Nash-Moser technique. An application to the nonlinear system of conservation law is done. Namely, simple sufficient conditions implying that for any \(d>-1\) the compatible initial data have index d are given.
Reviewer: V.Georgiev

MSC:
35L60 First-order nonlinear hyperbolic equations
35F25 Initial value problems for nonlinear first-order PDEs
35L67 Shocks and singularities for hyperbolic equations
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