On microlocal analyticity of solutions of first-order nonlinear PDE
Annales de l'Institut Fourier, Volume 59 (2009) no. 4, p. 1267-1290

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

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

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, σCharL u and 1 iσ([L u ,L u ¯])<0 or if u is a C 3 solution, σCharL u , σ([L u ,L u ¯])=0, and σ([L u ,[L u ,L u ¯]])0, then σWF a u. Here WF a u denotes the analytic wave-front set of u and CharL 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 L u ¯ of any order.

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

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

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 Char L u et 1 iσ([L u ,L u ¯])<0, ou si u est une solution de classe C 3 , σ Char L u , σ([L u ,L u ¯])=0 et σ([L u ,[L u ,L u ¯]])0, alors σWF a (u). Ici, WF 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 L u ¯ de tout ordre.

DOI : https://doi.org/10.5802/aif.2463
Classification:  35A18,  35B65,  35F20
Keywords: Analytic wave-front set, linearized operator
@article{AIF_2009__59_4_1267_0,
     author = {Berhanu, Shif},
     title = {On microlocal analyticity of solutions of first-order nonlinear PDE},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {4},
     year = {2009},
     pages = {1267-1290},
     doi = {10.5802/aif.2463},
     mrnumber = {2566960},
     zbl = {1195.35011},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_4_1267_0}
}
Berhanu, Shif. On microlocal analyticity of solutions of first-order nonlinear PDE. Annales de l'Institut Fourier, Volume 59 (2009) no. 4, pp. 1267-1290. doi : 10.5802/aif.2463. http://www.numdam.org/item/AIF_2009__59_4_1267_0/

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