On microlocal analyticity of solutions of first-order nonlinear PDE

Berhanu, Shif

Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1267-1290.

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

$$u_t=f(x,t,u,u_x)$$

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 $C^2$, $\sigma \in \mathrm{Char}{L}^u$ et $\frac{1}{i}\sigma \left([L^u,\overline{L^u}]\right)<0$, ou si $u$ est une solution de classe $C^3$, $\sigma \in \mathrm{Char}{L}^u$, $\sigma \left([L^u,\overline{L^u}]\right)=0$ et $\sigma \left([L^u,[L^u,\overline{L^u}]]\right)\ne 0$, alors $\sigma \notin W{F}_a\left(u\right)$. Ici, $W{F}_a\left(u\right)$ désigne le front d’onde analytique de $u$ et $\mathrm{Char}{L}^u$ 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.

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

$$u_t=f(x,t,u,u_x)$$

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 $\frac{1}{i}\sigma \left([L^u,\overline{L^u}]\right)<0$ or if $u$ is a $C^3$ solution, $\sigma \in \text{Char}{L}^u$, $\sigma \left([L^u,\overline{L^u}]\right)=0$, and $\sigma \left([L^u,[L^u,\overline{L^u}]]\right)\ne 0$, then $\sigma \notin W{F}_{a}u$. Here $W{F}_{a}u$ 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.

Classification : 35A18, 35B65, 35F20
Keywords: Analytic wave-front set, linearized operator
Mot clés : front d’onde analytique, opérateur linéarisé

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