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

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

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
@article{CRMATH_2003__336_3_219_0,
     author = {Zhuangchu Luo and Hua Chen and Changgui Zhang},
     title = {On the summability of the formal solutions for some {PDEs} with irregular singularity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {219--224},
     publisher = {Elsevier},
     volume = {336},
     number = {3},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00023-2},
     language = {en},
}
TY  - JOUR
AU  - Zhuangchu Luo
AU  - Hua Chen
AU  - Changgui Zhang
TI  - On the summability of the formal solutions for some PDEs with irregular singularity
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 219
EP  - 224
VL  - 336
IS  - 3
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00023-2
LA  - en
ID  - CRMATH_2003__336_3_219_0
ER  - 
%0 Journal Article
%A Zhuangchu Luo
%A Hua Chen
%A Changgui Zhang
%T On the summability of the formal solutions for some PDEs with irregular singularity
%J Comptes Rendus. Mathématique
%D 2003
%P 219-224
%V 336
%N 3
%I Elsevier
%R 10.1016/S1631-073X(03)00023-2
%G en
%F CRMATH_2003__336_3_219_0
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/

[1] W. Balser Formal power series and linear systems of meromorphic ordinary differential equations, Universitext, XVIII, Springer-Verlag, New York, 2000

[2] B.L.J. Braaksma Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier, Volume 42 (1992), pp. 517-540

[3] H. Chen, Z. Luo, R. Schäfke, C. Zhang, Summability of the formal solutions of some PDEs with irregular singularity, in preparation

[4] H. Chen; Z. Luo; H. Tahara Formal solution of nonlinear first order totally characteristic type PDE with irregular singularity, Ann. Inst. Fourier, Volume 51 (2001), pp. 1599-1620

[5] O. Costin; S. Tanveer Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane, Comm. Pure Appl. Math., Volume LIII (2000), pp. 0001-0026

[6] R. Gérard; H. Tahara Singular Nonlinear Partial Differential Equations, Aspects of Math., E 28, Vieweg, 1996

[7] D.A. Lutz; M. Miyake; R. Schäfke On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J., Volume 154 (1999), pp. 1-29

[8] J.P. Ramis Les séries k-sommables et leurs applications, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Phys., 126, 1980, pp. 178-199

Cité par Sources :

Commentaires - Politique