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Geometric explosion for quasilinear systems. (Explosion géométrique pour des systèmes quasi-linéaires.) (French) Zbl 0840.35060

A local geometric theory for blow-up in general quasilinear hyperbolic systems is sketched. The main idea of this theory is to correspond to a given system \(L\) another “blow-up” system \(Le\) and to correspond to each convenient solution of \(Le\) one blow-up solution of \(L\). This solution has the form \(u(\phi(x))= v(x)\), where \(\phi(x)= (\varphi(x), x_2,\dots, x_n)\) and \(u(\varphi, v)\) is a smooth solution of \(Le\). Of course, \(\partial\varphi(0)/\partial x_1 = 0\) and each singularities of \(\phi\) on \(x= 0\) (fold, crease etc…) corresponds to a solution with the same name. It is proved that there is no obstruction in \(Le\) to find solutions \((\varphi, v)\) the jet of which has the desired properties. In fact, this theory is microlocal in sense that it involves only on real eigenvalue of the system \(L\) in a characteristic point \((x^0, u^0, \varsigma^0)\). In more simple cases (folds, creases) it is analyzed the characteristics linearized on the constructed singular solution. That makes obvious the essential difference between one singular solution with the other blow-up solution. The blow-up solution is a solution which becomes singular in a point of a influence zone where it is regular: the observed singularity is therefore created, but not propagated. The link between the blow-up solutions and genuine nonlinearity is studied.
Reviewer: L.G.Vulkov (Russe)

MSC:

35L60 First-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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