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Interaction d’ondes simples pour des équations complètement non- linéaires. (Simple wave interaction for completely nonlinear equations). (French) Zbl 0608.35041
Sémin., Équations Dériv. Partielles 1985-1986, Exposé No. 8, 11 p. (1986).
The paper deals with quasi-linear or fully non-linear equations of hyperbolic type in a domain of \({\mathbb{R}}_ t\times {\mathbb{R}}^ n_ x\) (any n).
The main result describes the singularities of a given real solution u \((u\in H^{s+m}_{loc},s>(n+1)/2)\) of \[ F(x,t,u,...,\partial^{\alpha}u,...)=0\quad (| \alpha | \leq m), \] when the Cauchy data are assumed to be conormal with respect to some \(C^{\infty}\) hypersurface \(\Gamma\) (0\(\in \Gamma)\) contained in \(\{t=0\}\). If the linearized operator of F is assumed to be strictly hyperbolic, \(s>(n/2)+5\) and \(\partial^ j_ tu|_{t=0}\in H^{s+m-j,\infty}(\Gamma)\), u is shown to be \(C^{\infty}\) outside the characteristic surfaces \(\Sigma_ 1,...,\Sigma_ m\) through \(\Gamma\). Moreover, these surfaces are \(C^{\infty}\) outside \(\Gamma\), and \(u\in H^{s+m,\infty}(\Sigma_ j)\) near each point of \(\Sigma_ j\setminus \Gamma.\)
The case of two progressing waves, conormal with respect to disjoint smooth characteristic surfaces \(\Sigma_ 1\) and \(\Sigma_ 2\) in the past \(\{t<0\}\), meeting along \(\Gamma\) in the future \(\{\) \(t\geq 0\}\), is also handled: \(\Gamma\) is shown to be \(C^{\infty}\), and so is u outside \(\Sigma_ 1,\Sigma_ 2,\Sigma^+_ 3,...,\Sigma^+_ m\), where the \(\Sigma^+_ j\) are the outgoing characteristic surfaces from \(\Gamma\) (assumed to exist). Conormal regularity is also obtained in this case, as in the Cauchy problem.
This result seems to be the first general result on \(C^{\infty}\) singularities of solutions to (hyperbolic) quasi-linear or fully non- linear equations.
The method of proof relies on the construction of appropriate ”exotic” algebras of conormal distributions, using Bony’s paradifferential calculus.

35L75 Higher-order nonlinear hyperbolic equations
35L67 Shocks and singularities for hyperbolic equations
35L30 Initial value problems for higher-order hyperbolic equations
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