Comptes Rendus
Partial Differential Equations
Existence of local strong solutions for a quasilinear Benney system
[Existence d'une solution locale forte pour un système de Benney quasilinéaire]
Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 493-496.

We prove in this Note the existence and uniqueness of a strong local solution to the Cauchy problem for the quasilinear Benney system.

Nous prouvons dans cette Note l'existence et unicité d'une solution locale forte du problème de Cauchy pour le système de Benney quasilinéaire.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.03.005

João-Paulo Dias 1 ; Mário Figueira 1 ; Filipe Oliveira 2

1 CMAF/UL and FCUL, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
2 Dep. Matemática, FCT/UNL, Monte da Caparica, Portugal
@article{CRMATH_2007__344_8_493_0,
     author = {Jo\~ao-Paulo Dias and M\'ario Figueira and Filipe Oliveira},
     title = {Existence of local strong solutions for a quasilinear {Benney} system},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {493--496},
     publisher = {Elsevier},
     volume = {344},
     number = {8},
     year = {2007},
     doi = {10.1016/j.crma.2007.03.005},
     language = {en},
}
TY  - JOUR
AU  - João-Paulo Dias
AU  - Mário Figueira
AU  - Filipe Oliveira
TI  - Existence of local strong solutions for a quasilinear Benney system
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 493
EP  - 496
VL  - 344
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crma.2007.03.005
LA  - en
ID  - CRMATH_2007__344_8_493_0
ER  - 
%0 Journal Article
%A João-Paulo Dias
%A Mário Figueira
%A Filipe Oliveira
%T Existence of local strong solutions for a quasilinear Benney system
%J Comptes Rendus. Mathématique
%D 2007
%P 493-496
%V 344
%N 8
%I Elsevier
%R 10.1016/j.crma.2007.03.005
%G en
%F CRMATH_2007__344_8_493_0
João-Paulo Dias; Mário Figueira; Filipe Oliveira. Existence of local strong solutions for a quasilinear Benney system. Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 493-496. doi : 10.1016/j.crma.2007.03.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.03.005/

[1] D.J. Benney A general theory for interactions between short and long waves, Stud. Appl. Math., Volume 56 (1977), pp. 81-94

[2] J.P. Dias, M. Figueira, Existence of weak solutions for a quasilinear version of Benney equations, J. Hyperbolic Differential Equations, in press

[3] T. Kato Quasi-linear equations of evolution, with applications to partial differential equations, Lecture Notes in Math., vol. 448, Springer, 1975, pp. 25-70

[4] F. Oliveira Stability of the solitons for the one-dimensional Zakharov–Rubenchik equation, Physica D, Volume 175 (2003), pp. 220-240

[5] Y. Shibata; Y. Tsutsumi Local existence of solutions for the initial boundary problem of fully nonlinear wave equation, Nonlinear Anal. TMA, Volume 11 (1987), pp. 335-365

[6] M. Tsutsumi; S. Hatano Well-posedness of the Cauchy problem for the long wave–short wave resonance equations, Nonlinear Anal. TMA, Volume 22 (1994), pp. 155-171

[7] M. Tsutsumi; S. Hatano Well-posedness of the Cauchy problem for Benney's first equations of long wave short wave interactions, Funkcial. Ekvac., Volume 37 (1994), pp. 289-316

  • João-Paulo Dias A quasilinear system related with the asymptotic equation of the nematic liquid crystal's director field, Chinese Annals of Mathematics. Series B, Volume 42 (2021) no. 2, pp. 163-172 | DOI:10.1007/s11401-021-0251-z | Zbl:1475.35267
  • Jincheng Shi; Shengzhong Xiao Global existence and blow-up for the classical solutions of the long-short wave equations with viscosity, Discrete Dynamics in Nature and Society, Volume 2021 (2021), p. 16 (Id/No 7211126) | DOI:10.1155/2021/7211126 | Zbl:1486.35062
  • João-Paulo Dias; Filipe Oliveira; Hugo Tavares On a coupled system of a Ginzburg-Landau equation with a quasilinear conservation law, Communications in Contemporary Mathematics, Volume 22 (2020) no. 7, p. 30 (Id/No 1950054) | DOI:10.1142/s0219199719500548 | Zbl:1446.35191
  • João-Paulo Dias Linear stability of shock profiles for a quasilinear Benney system in \mathbb{R}^2{{\times}} {{\mathbb}}{R}_+, Journal of Hyperbolic Differential Equations, Volume 17 (2020) no. 4, pp. 797-807 | DOI:10.1142/s0219891620500253 | Zbl:1471.35031
  • João-Paulo Dias; Pedro Freitas Decay of solutions for a class of nonlinear Schrödinger equations in R and the stability of shock profiles for a quasilinear Benney system, Nonlinearity, Volume 31 (2018) no. 3, pp. 1110-1119 | DOI:10.1088/1361-6544/aaaa09 | Zbl:1390.35322
  • João-Paulo Dias; Filipe Oliveira On a quasilinear nonlocal Benney system, Journal of Hyperbolic Differential Equations, Volume 14 (2017) no. 1, pp. 135-156 | DOI:10.1142/s0219891617500047 | Zbl:1387.35562
  • Paulo Amorim; João-Paulo Dias; Mário Figueira; Philippe G. LeFloch The linear stability of shock waves for the nonlinear Schrödinger-inviscid Burgers system, Journal of Dynamics and Differential Equations, Volume 25 (2013) no. 1, pp. 49-69 | DOI:10.1007/s10884-012-9283-0 | Zbl:1307.35271
  • Paulo Amorim; João Paulo Dias A nonlinear model describing a short wave – long wave interaction in a viscoelastic medium, Quarterly of Applied Mathematics, Volume 71 (2013) no. 3, pp. 417-432 | DOI:10.1090/s0033-569x-2012-01298-4 | Zbl:1275.35082
  • João-Paulo Dias; Mário Figueira; Filipe Oliveira On the Cauchy problem describing an electron-phonon interaction, Chinese Annals of Mathematics. Series B, Volume 32 (2011) no. 4, pp. 483-496 | DOI:10.1007/s11401-011-0663-2 | Zbl:1347.37128
  • S. Antontsev; J. P. Dias; M. Figueira; F. Oliveira Non-existence of global solutions for a quasilinear Benney system, Journal of Mathematical Fluid Mechanics, Volume 13 (2011) no. 2, pp. 213-222 | DOI:10.1007/s00021-009-0014-1 | Zbl:1270.35358
  • S. Antontsev; J. P. Dias; M. Figueira; F. Oliveira Erratum to: “Non-existence of global solutions for a quasilinear Benney system”, Journal of Mathematical Fluid Mechanics, Volume 13 (2011) no. 2, p. 307 | DOI:10.1007/s00021-009-0021-2 | Zbl:1270.35359
  • João Paulo Dias; Mário Figueira; Hermano Frid Vanishing viscosity with short wave-long wave interactions for systems of conservation laws, Archive for Rational Mechanics and Analysis, Volume 196 (2010) no. 3, pp. 981-1010 | DOI:10.1007/s00205-009-0273-2 | Zbl:1203.35020
  • João-Paulo Dias; Mário Figueira; Filipe Oliveira Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 73 (2010) no. 8, pp. 2686-2698 | DOI:10.1016/j.na.2010.06.049 | Zbl:1194.35405

Cité par 13 documents. Sources : zbMATH

Commentaires - Politique