Comptes Rendus
no. 5
Partial differential equations
Almost global well-posedness of Kirchhoff equation with Gevrey data
[L'équation de Kirchhoff avec données de Gevrey est presque globalement bien posée]

Présenté par : the Editorial Board

Tokio Matsuyama1 ; Michael Ruzhansky2
1 Department of Mathematics, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
2 Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom
Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 522-525.

Le propos de cette Note est d'énoncer que l'équation de Kirchhoff avec des données grandes dans les espaces de Gevrey est presque globalement bien posée. Nous discutons aussi brièvement les résultats correspondants dans les domaines bornés et les domaines extérieurs.

The aim of this note is to present the almost global well-posedness result for the Cauchy problem for the Kirchhoff equation with large data in Gevrey spaces. We also briefly discuss the corresponding results in bounded and in exterior domains.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.04.001
Tokio Matsuyama 1 ; Michael Ruzhansky 2

1 Department of Mathematics, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
2 Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom 
@article{CRMATH_2017__355_5_522_0,
     author = {Tokio Matsuyama and Michael Ruzhansky},
     title = {Almost global well-posedness of {Kirchhoff} equation with {Gevrey} data},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {522--525},
     publisher = {Elsevier},
     volume = {355},
     number = {5},
     year = {2017},
     doi = {10.1016/j.crma.2017.04.001},
     language = {en},
}
TY  - JOUR
AU  - Tokio Matsuyama
AU  - Michael Ruzhansky
TI  - Almost global well-posedness of Kirchhoff equation with Gevrey data
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 522
EP  - 525
VL  - 355
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2017.04.001
LA  - en
ID  - CRMATH_2017__355_5_522_0
ER  - 
%0 Journal Article
%A Tokio Matsuyama
%A Michael Ruzhansky
%T Almost global well-posedness of Kirchhoff equation with Gevrey data
%J Comptes Rendus. Mathématique
%D 2017
%P 522-525
%V 355
%N 5
%I Elsevier
%R 10.1016/j.crma.2017.04.001
%G en
%F CRMATH_2017__355_5_522_0
Tokio Matsuyama; Michael Ruzhansky. Almost global well-posedness of Kirchhoff equation with Gevrey data. Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 522-525. doi : 10.1016/j.crma.2017.04.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.04.001/

[1] A. Arosio; S. Spagnolo Global solutions to the Cauchy problem for a nonlinear hyperbolic equation, Collège de France Seminar, vol. VI, Paris, 1982/1983 (H. Brézis; J.-L. Lions, eds.) (Research Notes in Mathematics), Volume vol. 109, Pitman, Boston, MA, USA (1984), pp. 1-26

[2] S. Bernstein Sur une classe d'équations fonctionnelles aux dérivées partielles, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 4 (1940), pp. 17-27

[3] F. Colombini; D. Del Santo; T. Kinoshita Well-posedness of a hyperbolic equation with non-Lipschitz coefficients, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 1 (2002), pp. 327-358

[4] P. D'Ancona; S. Spagnolo Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., Volume 108 (1992), pp. 247-262

[5] M. Ghisi; M. Gobbino Derivative loss for Kirchhoff equations with non-Lipschitz nonlinear term, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 8 (2009) no. 4, pp. 613-646

[6] M. Ghisi; M. Gobbino Kirchhoff equations in generalized Gevrey spaces: local existence, global existence, uniqueness, Rend. Ist. Mat. Univ. Trieste, Volume 42 (2010) no. supp., pp. 89-110

[7] M. Ghisi; M. Gobbino Kirchhoff equation from quasi-analytic to spectral-gap data, Bull. Lond. Math. Soc., Volume 43 (2011), pp. 374-385

[8] F. Hirosawa Global solvability for Kirchhoff equation in special classes of non-analytic functions, J. Differ. Equ., Volume 230 (2006), pp. 49-70

[9] K. Kajitani; K. Yamaguti On global analytic solutions of the degenerate Kirchhoff equation, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 21 (1994), pp. 279-297

[10] G. Kirchhoff Vorlesungen über Mechanik, Teubner, Leipzig, 1876

[11] R. Manfrin On the global solvability of Kirchhoff equation for non-analytic initial data, J. Differ. Equ., Volume 211 (2005), pp. 38-60

[12] T. Matsuyama; M. Ruzhansky Global well-posedness of Kirchhoff systems, J. Math. Pures Appl., Volume 100 (2013), pp. 220-240

[13] T. Matsuyama; M. Ruzhansky Global well-posedness of the Kirchhoff equation and Kirchhoff systems, Analytic Methods in Interdisciplinary Applications, Springer Proc. Math. Stat., vol. 116, Springer, 2015

[14] T. Matsuyama; M. Ruzhansky On the Gevrey well-posedness of the Kirchhoff equation, J. Anal. Math. (2017) (in press in 2020) | arXiv

[15] K. Nishihara On a global solution of some quasilinear hyperbolic equation, Tokyo J. Math., Volume 7 (1984), pp. 437-459

[16] S.I. Pohožhaev On a class of quasilinear hyperbolic equations, Math. USSR Sb., Volume 25 (1975), pp. 145-158

Cité par Sources :

Commentaires - Politique