Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations

[Des normes de Nagumo de type exponentiel et la sommabilité des solutions formelles d’équations singulières aux dérivées partielles]

Luo, Zhuangchu ; Chen, Hua ; Zhang, Changgui

Annales de l'Institut Fourier, Tome 62 (2012), p. 571-618 / Harvested from Numdam

Résumé

Dans cet article, nous étudions une classe d’équations aux dérivées partielles du premier ordre, non linéaires, dégénérées et ayant une singularité en $(t,x)=(0,0)\in \mathbf{C}^2$. Au moyen d’une famille de normes de Nagumo de type exponentiel, l’analyse asymptotique Gevrey s’étend naturellement au cas de paramètres holomorphes. Une condition optimale est ainsi établie pour déduire la $k$-sommabilité des solutions formelles. En outre, des solutions analytiques dans des domaines coniques sont obtenues pour chaque type de ces PDE singulières non linéaires.

Abstract

In this paper, we study a class of first order nonlinear degenerate partial differential equations with singularity at $(t,x)=(0,0)\in \mathbf{C}^2$. Using exponential-type Nagumo norm approach, the Gevrey asymptotic analysis is extended to case of holomorphic parameters in a natural way. A sharp condition is then established to deduce the $k$-summability of the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.

Détail

Publié le : 2012-01-01

MR 2985510 | Zbl 1252.30025

DOI : https://doi.org/10.5802/aif.2688
Classification:  30E15,  32D15,  35C10,  35C20
Mots clés: Norme Nagumo, équations différentielles singulières, singularité du type fuchsien, sommabilité de Borel, phénomène de Stokes, $k$-sommabilité, paramètres holomorphes.

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