On the summability of the formal solutions for some PDEs with irregular singularity

Zhuangchu Luo; Hua Chen; Changgui Zhang

doi:10.1016/S1631-073X(03)00023-2

Partial Differential Equations

On the summability of the formal solutions for some PDEs with irregular singularity
[Sur la sommabilité des solutions formelles de certaines EDP à singularité irrégulière]

Présenté par : Bernard Malgrange

Zhuangchu Luo ¹ ; Hua Chen ¹ ; Changgui Zhang ²

¹ Institute of Mathematics, Wuhan University, Wuhan 430072, PR China
² Laboratoire AGAT (UMR-CNRS 8524), UFR math., Université de Lille 1, cité scientifique, 59655 Villeneuve d'Ascq cedex, France

Comptes Rendus. Mathématique, Volume 336 (2003) no. 3, pp. 219-224.

Résumé
Abstract

Dans la présente Note, nous considérons des classes d'équations aux dérivées partielles, non linéaires et qui sont toutes singulières régulières en t=0 et irrégulières en x=0. Notre but est d'établir un résultat similaire à la k-sommabilité connue pour des équations différentielles méromorphes à points singuliers. Nous montrons que, sous certaines conditions de généricité, toutes les solutions formelles sont Borel sommables ou k-sommables dans toutes les directions du plan des x sauf éventuellement un nombre dénombrable.

In this Note, we consider some classes of nonlinear partial differential equations with regular singularity with respect to t=0 and irregular one with respect to x=0. Our purpose is to establish a result which is similar to the k-summability property, known in the case of singular ordinary differential equations. We can prove that, except at most a countable set, the formal solution is Borel summable or k-summable with respect to x in all other directions.

Accepté le : 2003-01-08
Publié le : 2003-02-01

DOI : 10.1016/S1631-073X(03)00023-2

Affiliations des auteurs :

Zhuangchu Luo ¹ ; Hua Chen ¹ ; Changgui Zhang ²

¹ Institute of Mathematics, Wuhan University, Wuhan 430072, PR China
² Laboratoire AGAT (UMR-CNRS 8524), UFR math., Université de Lille 1, cité scientifique, 59655 Villeneuve d'Ascq cedex, France

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     author = {Zhuangchu Luo and Hua Chen and Changgui Zhang},
     title = {On the summability of the formal solutions for some {PDEs} with irregular singularity},
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Zhuangchu Luo; Hua Chen; Changgui Zhang. On the summability of the formal solutions for some PDEs with irregular singularity. Comptes Rendus. Mathématique, Volume 336 (2003) no. 3, pp. 219-224. doi : 10.1016/S1631-073X(03)00023-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00023-2/

Bibliographie
Cité par

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