## Paracomposition et application aux équations non-linéaires.(French)Zbl 0578.35010

Sémin. Bony-Sjöstrand-Meyer, Équations Dériv. Partielles 1984-1985, Exp. No. 11, 10 p. (1985).
Let $$\chi$$ : $$\Omega$$ $${}_ 1\to \Omega_ 2$$ be a diffeomorphism of class $$C^{\rho +1}_{loc}$$ $$(\rho >0)$$ of two open sets in $${\mathbb{R}}^ n$$. The author announces several results concerning ”paracomposition” $$\chi^*u$$, $$u\in C_ 0^{\infty}(\Omega_ 2)'$$. Among others he states the following properties of $$\chi^*: C_ 0^{\infty}(\Omega_ 2)'\to C_ 0^{\infty}(\Omega_ 1)',$$
(i) $$\chi^*: H^ s_{loc}(\Omega_ 2)\to H^ s_{loc}(\Omega_ 1)$$, $$s\in {\mathbb{R}},$$
(ii) if $$\chi_ 0: \Omega_ 0\to \Omega_ 1$$ and $$\chi_ 1: \Omega_ 1\to \Omega_ 2$$ are two diffeomorphisms of class $$C^{\rho +1}$$ $$(\rho >0)$$ then $$\chi^*_ 0\chi^*_ 1u=(\chi_ 2\chi_ 0)^*u+Ru$$, where $$R: H^ s_{loc}(\Omega_ 2)\to H^{s+\rho}_{loc}(\Omega_ 1)$$, $$s\in {\mathbb{R}},$$
(iii) if $$h(x,\xi)\in \Sigma^ m_{\alpha}(\Omega_ 2)$$ (the class of symbols of Bony) $$\epsilon =\min (\alpha,\rho)$$, and $$T_ h$$ denotes the operator of Bony with symbol h then $\chi^*T_ hu=T_{h^*}\chi^*u+Ru,\quad h\in \Sigma^ m_{\epsilon}(\Omega_ 1),\quad R: H^ s_{loc}(\Omega_ 2)\to H_{loc}^{s- m+\epsilon}(\Omega_ 1).$ An application of the above results to the problem of regularity of the solution of the general nonlinear equation $$F(y,u,...,\partial^{\alpha}u,...)=0,$$ $$| \alpha | \leq m$$, F is real of $$C^{\infty}$$ class, is given, under certain additional assumption (hyperbolicity, and character conormal of solution in the past).
Reviewer: J.Janas

### MSC:

 35G20 Nonlinear higher-order PDEs 35L70 Second-order nonlinear hyperbolic equations 35B65 Smoothness and regularity of solutions to PDEs

### Keywords:

paracomposition; symbols of Bony; operator of Bony; regularity
Full Text: