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]
Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 571-618.

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)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.

In this paper, we study a class of first order nonlinear degenerate partial differential equations with singularity at (t,x)=(0,0)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.

DOI : 10.5802/aif.2688
Classification : 30E15, 32D15, 35C10, 35C20
Keywords: Nagumo norm, singular differential equations, Fuchsian singularity, Borel summability, Stokes phenomenon, $k$-summability, holomorphic parameters.
Mot clés : Norme Nagumo, équations différentielles singulières, singularité du type fuchsien, sommabilité de Borel, phénomène de Stokes, $k$-sommabilité, paramètres holomorphes.
Luo, Zhuangchu 1 ; Chen, Hua 1 ; Zhang, Changgui 2

1 Wuhan University, School of Mathematics and Statistics, Wuhan 430072, China
2 Université de Lille 1, Laboratoire P. Painlevé (UMR–CNRS 8524), UFR Math., Cité scientifique, 59655 Villeneuve d’Ascq cedex, France
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Luo, Zhuangchu; Chen, Hua; Zhang, Changgui. Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations. Annales de l'Institut Fourier, Tome 62 (2012) no. 2, pp. 571-618. doi : 10.5802/aif.2688. http://www.numdam.org/articles/10.5802/aif.2688/

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