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Zbl 0608.35041
Alinhac, S.
Interaction d'ondes simples pour des équations complètement non- linéaires. (Simple wave interaction for completely nonlinear equations). (French)
[J] 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 ${\bbfR}\sb t\times {\bbfR}\sp n\sb x$ (any n). \par The main result describes the singularities of a given real solution u $(u\in H\sp{s+m}\sb{loc},s>(n+1)/2)$ of $$ F(x,t,u,...,\partial\sp{\alpha}u,...)=0\quad (\vert \alpha \vert \le m), $$ when the Cauchy data are assumed to be conormal with respect to some $C\sp{\infty}$ 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 $\partial\sp j\sb tu\vert\sb{t=0}\in H\sp{s+m-j,\infty}(\Gamma)$, u is shown to be $C\sp{\infty}$ outside the characteristic surfaces $\Sigma\sb 1,...,\Sigma\sb m$ through $\Gamma$. Moreover, these surfaces are $C\sp{\infty}$ outside $\Gamma$, and $u\in H\sp{s+m,\infty}(\Sigma\sb j)$ near each point of $\Sigma\sb j\setminus \Gamma.$ \par The case of two progressing waves, conormal with respect to disjoint smooth characteristic surfaces $\Sigma\sb 1$ and $\Sigma\sb 2$ in the past $\{t<0\}$, meeting along $\Gamma$ in the future $\{$ $t\ge 0\}$, is also handled: $\Gamma$ is shown to be $C\sp{\infty}$, and so is u outside $\Sigma\sb 1,\Sigma\sb 2,\Sigma\sp+\sb 3,...,\Sigma\sp+\sb m$, where the $\Sigma\sp+\sb j$ are the outgoing characteristic surfaces from $\Gamma$ (assumed to exist). Conormal regularity is also obtained in this case, as in the Cauchy problem. \par This result seems to be the first general result on $C\sp{\infty}$ singularities of solutions to (hyperbolic) quasi-linear or fully non- linear equations. \par The method of proof relies on the construction of appropriate "exotic" algebras of conormal distributions, using Bony's paradifferential calculus.

MSC 2000:: *35L75 Nonlinear hyperbolic PDE of higher $(>2)$ order
35L67 Shocks, etc.
35L30 Higher order hyperbolic equations, initial value problems

Keywords: interaction; fully non-linear equations of hyperbolic type; singularities; Cauchy data; linearized operator; progressing waves; characteristic surfaces; Conormal regularity; quasi-linear; conormal distributions; Bony's paradifferential calculus

Cited in: Zbl 0744.35020

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Zentralblatt MATH Berlin [Germany]