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

Alinhac, S.

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