[Sur la sommabilité de séries entières divergentes satisfaisant des équations aux dérivées partielles avec singularités]
Le but de cette Note est d'appliquer la méthode de sommation de Borel–Laplace étudiée par H. Chen, Z. Luo et C. Zhang (Summability of formal solutions of singular PDEs by means of two-dimensional Borel–Laplace method, preprint) aux solutions séries entières de deux familles d'EDP non linéaires. La première contient particulièrement une version bidimensionnelle de ce qu'on appelle équation d'Éuler, alors que la seconde famille d'EDP est dite de type totalement caractéristique par H. Chen et H. Tahara (On the holomorphic solution of non-linear totally characteristic equations, Math. Nachr. 219 (2000) 85–96).
The aim of this note is to apply the Borel–Laplace summation method studied by H. Chen, Z. Luo and C. Zhang (Summability of formal solutions of singular PDEs by means of two-dimensional Borel–Laplace method, preprint) to the divergent power series solutions to two families of nonlinear PDEs. The first one contains particularly a two-dimensional version of the so-called Euler equation (ODE), while the second is called totally characteristic type PDE by H. Chen and H. Tahara (On the holomorphic solution of non-linear totally characteristic equations, Math. Nachr. 219 (2000) 85–96).
Accepté le :
Publié le :
@article{CRMATH_2019__357_3_258_0,
author = {Chen, Hua and Luo, Zhuangchu and Zhang, Changgui},
title = {On the summability of divergent power series satisfying singular {PDEs}},
journal = {Comptes Rendus. Math\'ematique},
pages = {258--262},
publisher = {Elsevier},
volume = {357},
number = {3},
year = {2019},
doi = {10.1016/j.crma.2019.02.008},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2019.02.008/}
}
TY - JOUR AU - Chen, Hua AU - Luo, Zhuangchu AU - Zhang, Changgui TI - On the summability of divergent power series satisfying singular PDEs JO - Comptes Rendus. Mathématique PY - 2019 SP - 258 EP - 262 VL - 357 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2019.02.008/ DO - 10.1016/j.crma.2019.02.008 LA - en ID - CRMATH_2019__357_3_258_0 ER -
%0 Journal Article %A Chen, Hua %A Luo, Zhuangchu %A Zhang, Changgui %T On the summability of divergent power series satisfying singular PDEs %J Comptes Rendus. Mathématique %D 2019 %P 258-262 %V 357 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2019.02.008/ %R 10.1016/j.crma.2019.02.008 %G en %F CRMATH_2019__357_3_258_0
Chen, Hua; Luo, Zhuangchu; Zhang, Changgui. On the summability of divergent power series satisfying singular PDEs. Comptes Rendus. Mathématique, Tome 357 (2019) no. 3, pp. 258-262. doi : 10.1016/j.crma.2019.02.008. http://www.numdam.org/articles/10.1016/j.crma.2019.02.008/
[1] Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Universitext, Springer-Verlag, New York, 2000
[2] Cauchy problems with characteristic initial hypersurface, Commun. Pure Appl. Math., Volume 26 (1983), pp. 455-475
[3] Éléments de mathématique, fasc. XXXIII, Variétés différentielles et analytiques, Actualités scientifiques et industrielles, vol. 1333, Hermann, Paris, 1967 Fascicule de résultats (Paragraphes 1 à 7)
[4] An extension of Borel–Laplace methods and monomial summability, J. Math. Anal. Appl., Volume 457 (2018) no. 1, pp. 461-477
[5] Formal solutions for first-order nonlinear PDEs with irregular singularity, J. Math. (Wuhan), Volume 32 (2012) no. 4, pp. 729-739
[6] H. Chen, Z. Luo, C. Zhang, Summability of formal solutions of singular PDEs by means of two-dimensional Borel–Laplace method, preprint.
[7] On the holomorphic solution of non-linear totally characteristic equations, Math. Nachr., Volume 219 (2000), pp. 85-96
[8] Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane, Commun. Pure Appl. Math., Volume LIII (2000)
[9] Singular Nonliear Partial Differential Equations, Aspects of Mathematics, E, vol. 28, Vieweg, 1996
[10] Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations, Ann. Inst. Fourier (Grenoble), Volume 62 (2012) no. 2, pp. 571-618
[11] On a family of symmetric hypergeometric functions of several variables and their Euler type integral representation, Adv. Math., Volume 252 (2014), pp. 652-683
[12] Asymptotic Analysis for Integrable Connections with Irregular Singular Points, Lecture Notes in Mathematics, vol. 1075, Springer-Verlag, Berlin, 1984
[13] Topological tensor products and asymptotic developments, Ann. Fac. Sci. Toulouse Math. (6), Volume 8 (1999) no. 2, pp. 281-295
[14] Gevrey asymptotics and applications to holomorphic ordinary differential equations, Differential Equations & Asymptotic Theory in Mathematical Physics, Ser. Anal., vol. 2, World Sci. Publ., Hackensack, NJ, USA, 2004, pp. 44-99
[15] Summability in a direction of formal power series in several variables, Asymptot. Anal., Volume 29 (2002) no. 2, pp. 115-141
[16] Fuchsian type equations and Fuchsian hyperbolic equations, Jpn. J. Math., Volume 5 (1979), pp. 245-347
[17] A new Taylor type formula and extensions for asymptotically developable functions, Stud. Math., Volume 123 (1997) no. 2, pp. 151-163
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☆ The research supported partially by the NSFC Grants (No. 11171261, 11371282 and 11631011).
