Partial differential equations
Local well-posedness of a nonlinear KdV-type equation
Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 895-899.

In this paper, a generalized nonlinear KdV equation with time- and space-dependent coefficients is considered. We show that the control of the dispersive and “diffusion” terms is possible if we use an adequate weight function determined with respect to the dispersive and “diffusion” coefficients to define the energy. We use the dispersive properties of the equation to prove the existence and uniqueness of solutions.

Dans cette note, on considère une équation de KdV généralisée avec coefficients variables en temps et en espace. On montre que les termes de « diffusion » et de dispersion peuvent être contrôlés en utilisant une fonction poids, déterminée en fonction des coefficients de « diffusion » et de dispersion, appropriée pour définir lʼénergie ; puis, en utilisant la propriété de dispersion de lʼéquation, on montre un résultat dʼexistence et dʼunicité des solutions.

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DOI: 10.1016/j.crma.2013.10.032
Israwi, Samer 1, 2; Talhouk, Raafat 2

1 Center for Research in Applied Mathematics and Statistics, Arts Sciences and Technology University in Lebanon (AUL), 113-7504 Beirut, Lebanon
2 Department and Laboratory of Mathematics, Faculty of Sciences 1, Doctoral School of Sciences and Technology, Lebanese University, Hadath, Lebanon
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Israwi, Samer; Talhouk, Raafat. Local well-posedness of a nonlinear KdV-type equation. Comptes Rendus. Mathématique, Volume 351 (2013) no. 23-24, pp. 895-899. doi : 10.1016/j.crma.2013.10.032. http://www.numdam.org/articles/10.1016/j.crma.2013.10.032/

[1] Alinhac, S.; Gérard, P. Opérateurs pseudo-différentiels et théorème de Nash–Moser, Savoirs ectuels, InterÉditions, Éditions du Centre national de la recherche scientifique, CNRS, Paris, Meudon, France, 1991 (190 p)

[2] T. Akhunov, A sharp condition for the well-posedness of the linear KdV-type equation, submitted for publication, , 11 Jan. 2013. | arXiv

[3] Craig, W.; Kappeler, T.; Strauss, W. Gain of regularity for equations of KdV type, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 9 (1992) no. 2, pp. 147-186

[4] Lannes, D. Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal., Volume 232 (2006), pp. 495-539

[5] Linares, F.; Ponce, G. Introduction to Nonlinear Dispersive Equations, Springer Science, Business Media, 2009

[6] Tian, Bo; Gao, Yi-Tian Variable-coefficient balancing-act method and variable-coefficient KdV equation from fluid dynamics and plasma physics, Eur. Phys. J. B, Volume 22 (2001), pp. 351-360

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