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

Zhuangchu Luo[1]; Hua Chen[2]; Changgui Zhang[3]

  • [1] Wuhan University, School of Mathematics and Statistics, Wuhan 430072, China
  • [2] Wuhan University, School of Mathematics and Statistics, Wuhan 430072, China
  • [3] 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 (2012)

  • Volume: 62, Issue: 2, page 571-618
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Luo, Zhuangchu, Chen, Hua, and Zhang, Changgui. "Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations." Annales de l’institut Fourier 62.2 (2012): 571-618. <http://eudml.org/doc/251139>.

@article{Luo2012,
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.},
affiliation = {Wuhan University, School of Mathematics and Statistics, Wuhan 430072, China; Wuhan University, School of Mathematics and Statistics, Wuhan 430072, China; Université de Lille 1, Laboratoire P. Painlevé (UMR–CNRS 8524), UFR Math., Cité scientifique, 59655 Villeneuve d’Ascq cedex, France},
author = {Luo, Zhuangchu, Chen, Hua, Zhang, Changgui},
journal = {Annales de l’institut Fourier},
keywords = {Nagumo norm; singular differential equations; Fuchsian singularity; Borel summability; Stokes phenomenon; $k$-summability; holomorphic parameters; -summability},
language = {eng},
number = {2},
pages = {571-618},
publisher = {Association des Annales de l’institut Fourier},
title = {Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations},
url = {http://eudml.org/doc/251139},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Luo, Zhuangchu
AU - Chen, Hua
AU - Zhang, Changgui
TI - Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 2
SP - 571
EP - 618
AB - 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.
LA - eng
KW - Nagumo norm; singular differential equations; Fuchsian singularity; Borel summability; Stokes phenomenon; $k$-summability; holomorphic parameters; -summability
UR - http://eudml.org/doc/251139
ER -

References

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