no. 2
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, Zhuangchu1 ; Chen, Hua1 ; Zhang, Changgui2
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
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2688/

[1] Balser, W. Formal power series and linear systems of meromorphic ordinary differential equations, Universitext XVIII, Springer-Verlag, New York, 2000 | MR

[2] Balser, W. Multisummability of formal power series solutions of partial differential equations with constant coefficients, J. Differential Equations, Volume 201 (2004), pp. 63-74 | DOI | MR

[3] Braaksma, B. L. J. Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier, Volume 42 (1992), pp. 517-540 | DOI | Numdam | MR | Zbl

[4] Canalis-Durand, M.; Ramis, J.-P.; Schäfke, R.; Sibuya, Y. Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., Volume 518 (2000), pp. 95-129 | DOI | MR

[5] Chen, H.; Luo, Z. On the holomorphic solution of non-linear totally characteristic equations with several space variables, Acta Mathematica Scientia, Volume 22B (2002), pp. 393-403 | MR

[6] Chen, H.; Luo, Z.; Tahara, H. Formal solution of nonlinear first order totally characteristic type PDE with irregular singularity, Ann. Inst. Fourier, Volume 51 (2001), pp. 1599-1620 | DOI | Numdam | MR

[7] Chen, H.; Luo, Z.; Zhang, C. On the summability of formal solutions for a class of nonlinear singular PDEs with irregular singularity, Contemporary of Mathematics, Volume 400 (2006), pp. 53-64 | MR

[8] Chen, H.; Tahara, H. On totally characteristic type non-linear differential equations in the Complex Domain, Publ. RIMS, Kyoto Univ., Volume 35 (1999), pp. 621-636 | DOI | MR | Zbl

[9] Chen, H.; Tahara, H. On the holomorphic solution of non-linear totally characteristic equations, Mathematische Nachrichten, Volume 219 (2000), pp. 85-96 | DOI | MR

[10] Costin, O.; Tanveer, S. Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane, Comm. Pure and Appl. Math., Volume LIII (2000), pp. 0001-0026 | MR

[11] Di Vizio, L. An ultrametric version of the Maillet-Malgrange theorem for non linear q-difference equations, Proc. Amer. Math. Soc., Volume 136 (2008), pp. 2803-2814 | DOI | MR

[12] Fife, P.C.; McLeod, J.B. The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., Volume 65 (1977), pp. 335-361 | DOI | MR | Zbl

[13] Gérard, R.; Tahara, H. Singular nonlinear partial differential equations, Aspects of Mathematics, E 28, Vieweg Verlag, 1996 | MR | Zbl

[14] Gevrey, M. Sur les équations aux dérivées partielles du type parabolique, J. de Mathématique, Volume 9 (1913), pp. 305-476

[15] Hagan, P. S.; Ockendon, J. R. Half-range analysis of a counter-current separator, J. Math. Anal. Appl., Volume 160 (1991), pp. 358-378 | DOI | MR | Zbl

[16] Hibino, M. Borel summability of divergent solutions for singular first order linear partial differential equations with polynomial coefficients, J. Math. Sci. Univ. Tokyo, Volume 10 (2003), pp. 279-309 | MR

[17] Hibino, M. Borel summability of divergence solutions for singular first-order partial differential equations with variable coefficients. I, J. Differential Equations, Volume 227 (2006), pp. 499-533 | DOI | MR

[18] Hibino, M. Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients. II, J. Differential Equations, Volume 227 (2006), pp. 534-563 | DOI | MR

[19] Hörmander, L. An introduction to complex analysis in several variables, North-Holland Mathematical Library, 7., North-Holland Publishing Co., Amsterdam, 1990 | MR | Zbl

[20] Luo, Z.; Chen, H.; Zhang, C. On the summability of the formal solutions for some PDEs with irregular singularity, C.R. Acad. Sci. Paris, Volume Sér. I, 336 (2003), pp. 219-224 | DOI | MR

[21] Luo, Z.; Zhang, C. On the Borel summability of divergent power series respective to two variables (Preprint, 2010)

[22] Lutz, D. A.; Miyake, M.; Schäfke, R. On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J., Volume 154 (1999), pp. 1-29 | MR | Zbl

[23] Malgrange, B. Sur le théorème de Maillet, Asymptot. Anal., Volume 2 (1989), pp. 1-4 | MR | Zbl

[24] Martinet, J.; Ramis, J.-P. Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Publ. Math., Inst. Hautes Études Sci., Volume 55 (1982), pp. 63-164 | DOI | Numdam | MR | Zbl

[25] Martinet, J.; Ramis, J.-P. Elementary acceleration and multisummability I, Annales de l’I.H.P. Physique théorique, Volume 54 (1991), pp. 331-401 | Numdam | MR | Zbl

[26] Nagumo, M. Über das Anfangswertproblem partieller Differentialgleichungen, Jap. J. Math., Volume 18 (1942), pp. 41-47 | MR | Zbl

[27] Ouchi, S. Multisummability of formal solutions of some linear partial differential equations, J. Differential Equations, Volume 185 (2002), pp. 513-549 | DOI | MR

[28] Ouchi, S. Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields, J. Math. Soc. Japan, Volume 57 (2005), pp. 415-460 | DOI | MR

[29] Ouchi, S. Multisummability of formal power series solutions of nonlinear partial differential equations in complex domains, Asymptot. Anal., Volume 47 (2006), pp. 187-225 | MR

[30] Pagani, C. D.; Talenti, G. On a forward-backward parabolic equation, Ann. Mat. Pura. Appl., Volume 90 (1971), pp. 1-57 | DOI | MR | Zbl

[31] Ramis, J.-P. Les séries k-sommables et leurs applications, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory (Lecture Notes in Physics), Volume 126 (1980), pp. 178-199 | MR

[32] Tougeron, J.-C. Sur les ensembles semi-analytiques avec conditions Gevrey au bord, Ann. Sci. École Norm. Sup., Volume 27 (1994), pp. 173-208 | Numdam | MR | Zbl

[33] Zhang, C. Sur un théorème du type de Maillet-Malgrange pour les équations q-différences-différentielles, Asymptot. Anal., Volume 17 (1998), pp. 309-314 | MR | Zbl

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