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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.

##### MSC:
 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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