On microlocal analyticity of solutions of first-order nonlinear PDE

Berhanu, Shif

doi:10.5802/aif.2463

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 ¹

¹ Temple University Department of Mathematics Philadelphia, PA 19122 (USA)

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

Résumé
Abstract

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

u_{t} = f (x, t, u, u_{x})

où $f (x, t, ζ_{0}, ζ)$ est une fonction analytique réelle, à valeurs complexes, et holomorphe en $(ζ_{0}, ζ)$ . Nous montrons que si $u$ est une solution de classe $C^{2}$ , $σ \in Char L^{u}$ et $\frac{1}{i} σ ([L^{u}, \bar{L^{u}}]) < 0$ , ou si $u$ est une solution de classe $C^{3}$ , $σ \in Char L^{u}$ , $σ ([L^{u}, \bar{L^{u}}]) = 0$ et $σ ([L^{u}, [L^{u}, \bar{L^{u}}]]) \neq 0$ , alors $σ \notin W F_{a} (u)$ . Ici, $W F_{a} (u)$ désigne le front d’onde analytique de $u$ et $Char L^{u}$ 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 $\bar{L^{u}}$ de tout ordre.

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

u_{t} = f (x, t, u, u_{x})

where $f (x, t, ζ_{0}, ζ)$ is complex-valued, real analytic in all its arguments and holomorphic in $(ζ_{0}, ζ)$ . We show that if the function $u$ is a $C^{2}$ solution, $σ \in Char L^{u}$ and $\frac{1}{i} σ ([L^{u}, \bar{L^{u}}]) < 0$ or if $u$ is a $C^{3}$ solution, $σ \in Char L^{u}$ , $σ ([L^{u}, \bar{L^{u}}]) = 0$ , and $σ ([L^{u}, [L^{u}, \bar{L^{u}}]]) \neq 0$ , then $σ \notin W F_{a} u$ . 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 $\bar{L^{u}}$ of any order.

EuDML MR Zbl

DOI : 10.5802/aif.2463

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

Affiliations des auteurs :

Berhanu, Shif ¹

¹ Temple University Department of Mathematics Philadelphia, PA 19122 (USA)

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     author = {Berhanu, Shif},
     title = {On microlocal analyticity of solutions of first-order nonlinear {PDE}},
     journal = {Annales de l'Institut Fourier},
     pages = {1267--1290},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {4},
     year = {2009},
     doi = {10.5802/aif.2463},
     mrnumber = {2566960},
     zbl = {1195.35011},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2463/}
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Berhanu, Shif. On microlocal analyticity of solutions of first-order nonlinear PDE. Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1267-1290. doi : 10.5802/aif.2463. https://aif.centre-mersenne.org/articles/10.5802/aif.2463/

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