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.

Reçu le : 2010-10-22
Accepté le : 2011-03-15
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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     author = {Luo, Zhuangchu and Chen, Hua and Zhang, Changgui},
     title = {Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations},
     journal = {Annales de l'Institut Fourier},
     pages = {571--618},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {2},
     year = {2012},
     doi = {10.5802/aif.2688},
     zbl = {1252.30025},
     mrnumber = {2985510},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2012__62_2_571_0/}
}
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/item/AIF_2012__62_2_571_0/

[1] Balser, W. Formal power series and linear systems of meromorphic ordinary differential equations, Universitext XVIII, Springer-Verlag, New York, 2000 | MR 1722871

[2] Balser, W. Multisummability of formal power series solutions of partial differential equations with constant coefficients, J. Differential Equations, Tome 201 (2004), pp. 63-74 | Article | MR 2057538

[3] Braaksma, B. L. J. Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier, Tome 42 (1992), pp. 517-540 | Article | Numdam | MR 1182640 | Zbl 0759.34003

[4] Canalis-Durand, M.; Ramis, J.-P.; Schäfke, R.; Sibuya, Y. Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., Tome 518 (2000), pp. 95-129 | Article | MR 1739408

[5] Chen, H.; Luo, Z. On the holomorphic solution of non-linear totally characteristic equations with several space variables, Acta Mathematica Scientia, Tome 22B (2002), pp. 393-403 | MR 1921319

[6] Chen, H.; Luo, Z.; Tahara, H. Formal solution of nonlinear first order totally characteristic type PDE with irregular singularity, Ann. Inst. Fourier, Tome 51 (2001), pp. 1599-1620 | Article | Numdam | MR 1871282

[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, Tome 400 (2006), pp. 53-64 | MR 2222465

[8] Chen, H.; Tahara, H. On totally characteristic type non-linear differential equations in the Complex Domain, Publ. RIMS, Kyoto Univ., Tome 35 (1999), pp. 621-636 | Article | MR 1719863 | Zbl 0961.35002

[9] Chen, H.; Tahara, H. On the holomorphic solution of non-linear totally characteristic equations, Mathematische Nachrichten, Tome 219 (2000), pp. 85-96 | Article | MR 1791913

[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., Tome LIII (2000), pp. 0001-0026 | MR 1761410

[11] Di Vizio, L. An ultrametric version of the Maillet-Malgrange theorem for non linear q-difference equations, Proc. Amer. Math. Soc., Tome 136 (2008), pp. 2803-2814 | Article | MR 2399044

[12] Fife, P.C.; McLeod, J.B. The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., Tome 65 (1977), pp. 335-361 | Article | MR 442480 | Zbl 0361.35035

[13] Gérard, R.; Tahara, H. Singular nonlinear partial differential equations, Aspects of Mathematics, E 28, Vieweg Verlag, 1996 | MR 1757086 | Zbl 0874.35001

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

[15] Hagan, P. S.; Ockendon, J. R. Half-range analysis of a counter-current separator, J. Math. Anal. Appl., Tome 160 (1991), pp. 358-378 | Article | MR 1126123 | Zbl 0753.76190

[16] Hibino, M. Borel summability of divergent solutions for singular first order linear partial differential equations with polynomial coefficients, J. Math. Sci. Univ. Tokyo, Tome 10 (2003), pp. 279-309 | MR 1987134

[17] Hibino, M. Borel summability of divergence solutions for singular first-order partial differential equations with variable coefficients. I, J. Differential Equations, Tome 227 (2006), pp. 499-533 | Article | MR 2237677

[18] Hibino, M. Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients. II, J. Differential Equations, Tome 227 (2006), pp. 534-563 | Article | MR 2237678

[19] Hörmander, L. An introduction to complex analysis in several variables, North-Holland Mathematical Library, 7., North-Holland Publishing Co., Amsterdam, 1990 | MR 1045639 | Zbl 0271.32001

[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, Tome Sér. I, 336 (2003), pp. 219-224 | Article | MR 1968262

[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., Tome 154 (1999), pp. 1-29 | MR 1689170 | Zbl 0958.35061

[23] Malgrange, B. Sur le théorème de Maillet, Asymptot. Anal., Tome 2 (1989), pp. 1-4 | MR 991413 | Zbl 0693.34004

[24] Martinet, J.; Ramis, J.-P. Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Publ. Math., Inst. Hautes Études Sci., Tome 55 (1982), pp. 63-164 | Article | Numdam | MR 672182 | Zbl 0546.58038

[25] Martinet, J.; Ramis, J.-P. Elementary acceleration and multisummability I, Annales de l’I.H.P. Physique théorique, Tome 54 (1991), pp. 331-401 | Numdam | MR 1128863 | Zbl 0748.12005

[26] Nagumo, M. Über das Anfangswertproblem partieller Differentialgleichungen, Jap. J. Math., Tome 18 (1942), pp. 41-47 | MR 15186 | Zbl 0061.21107

[27] Ouchi, S. Multisummability of formal solutions of some linear partial differential equations, J. Differential Equations, Tome 185 (2002), pp. 513-549 | Article | MR 1935612

[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, Tome 57 (2005), pp. 415-460 | Article | MR 2123239

[29] Ouchi, S. Multisummability of formal power series solutions of nonlinear partial differential equations in complex domains, Asymptot. Anal., Tome 47 (2006), pp. 187-225 | MR 2233920

[30] Pagani, C. D.; Talenti, G. On a forward-backward parabolic equation, Ann. Mat. Pura. Appl., Tome 90 (1971), pp. 1-57 | Article | MR 313635 | Zbl 0238.35043

[31] Ramis, J.-P. Les séries k-sommables et leurs applications, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory (Lecture Notes in Physics) Tome 126 (1980), pp. 178-199 | MR 579749

[32] Tougeron, J.-C. Sur les ensembles semi-analytiques avec conditions Gevrey au bord, Ann. Sci. École Norm. Sup., Tome 27 (1994), pp. 173-208 | Numdam | MR 1266469 | Zbl 0803.32003

[33] Zhang, C. Sur un théorème du type de Maillet-Malgrange pour les équations q-différences-différentielles, Asymptot. Anal., Tome 17 (1998), pp. 309-314 | MR 1656811 | Zbl 0938.34064