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

Luo, Zhuangchu; Chen, Hua; Zhang, Changgui

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 $(t,x)=(0,0)\in {\mathbf{C}}^2$. 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.

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.

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.

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