[Sur la sommabilité des solutions formelles de certaines EDP à singularité irrégulière]
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.
Publié le :
@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 -
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] Formal power series and linear systems of meromorphic ordinary differential equations, Universitext, XVIII, Springer-Verlag, New York, 2000
[2] 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] Formal solution of nonlinear first order totally characteristic type PDE with irregular singularity, Ann. Inst. Fourier, Volume 51 (2001), pp. 1599-1620
[5] 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] Singular Nonlinear Partial Differential Equations, Aspects of Math., E 28, Vieweg, 1996
[7] On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J., Volume 154 (1999), pp. 1-29
[8] 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
On summability of formal solutions to a Cauchy problem and generalization of Mordell's Theorem
Shuang Zhou; Zhuangchu Luo; Changgui Zhang
C. R. Math (2010)
On the summability of divergent power series satisfying singular PDEs
Hua Chen; Zhuangchu Luo; Changgui Zhang
C. R. Math (2019)
Colin Christopher; Christiane Rousseau
C. R. Math (2007)