On microlocal analyticity of solutions of first-order nonlinear PDE

[Sur l’analyticité microlocale des solutions d’équations aux dérivées partielles non linéaires du premier ordre]

Berhanu, Shif

Annales de l'Institut Fourier, Tome 59 (2009), p. 1267-1290 / Harvested from Numdam

Résumé

Nous étudions l’analyticité microlocale des solutions de l’équation non linéaire

ut=f(x,t,u,ux)

où $f(x,t,\zeta _0,\zeta )$ est une fonction analytique réelle, à valeurs complexes, et holomorphe en $(\zeta _0,\zeta )$. Nous montrons que si $u$ est une solution de classe $C^2$, $\sigma \in {\rm Char}\, L^u$ et $\frac{1}{i}\sigma ([L^u,\overline{L^u}])<0$, ou si $u$ est une solution de classe $C^3$, $\sigma \in {\rm Char}\, L^u$, $\sigma ([L^u,\overline{L^u}])=0$ et $\sigma ([L^u,[L^u,\overline{L^u}]])\ne 0$, alors $\sigma \notin WF_a(u)$. Ici, $WF_a(u)$ désigne le front d’onde analytique de $u$ et ${\rm Char}\, L^u$ l’ensemble caractéristique de l’opérateur linéarisé. Quand $m=1$, nous démontrons un résultat plus général faisant intervenir les crochets des opérateurs $L^u$ et $\overline{L^u}$ de tout ordre.

Abstract

We study the microlocal analyticity of solutions $u$ of the nonlinear equation

ut=f(x,t,u,ux)

where $f(x, t,\zeta _{0}, \zeta )$ is complex-valued, real analytic in all its arguments and holomorphic in $(\zeta _{0}, \zeta )$. We show that if the function $u$ is a $C^2$ solution, $\sigma \in \text{Char}\,L^{u}$ and $\frac{1}{i}\sigma ([L^{u}, \overline{L^u}]) < 0$ or if $u$ is a $C^3$ solution, $\sigma \in \text{Char}\,L^{u}$, $\sigma ([L^{u}, \overline{L^u}]) = 0$, and $ \sigma ([L^u,[L^{u},\overline{L^u}]]) \ne 0$, then $\sigma \notin WF_{a}u$. Here $WF_{a}u $ denotes the analytic wave-front set of $u$ and Char$\,L^u$ is the characteristic set of the linearized operator. When $m=1$, we prove a more general result involving the repeated brackets of $L^u$ and $\overline{L^u}$ of any order.

Détail

Publié le : 2009-01-01

MR 2566960 | Zbl 1195.35011

DOI : https://doi.org/10.5802/aif.2463
Classification:  35A18,  35B65,  35F20
Mots clés: front d’onde analytique, opérateur linéarisé

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