Abstract
Using Bourgain spaces and the generator of dilation P=3t ∂ t +x ∂ x , which almost commutes with the linear Korteweg-de Vries operator, we show that a solution of the initial value problem associated for the coupled system of equations of Korteweg-de Vries type which appears as a model to describe the strong interaction of weakly nonlinear long waves, has an analyticity in time and a smoothing effect up to real analyticity if the initial data only have a single point singularity at x=0.
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References
Alves, M., Villagrán, O. Vera: Smoothing properties for a coupled system of nonlinear evolution dispersive equations. Indag. Math. 20(2), 285–327 (2009)
Ash, J.M., Cohen, J., Wang, G.: On strongly interacting internal solitary waves. J. Fourier Anal. Appl. 2(5), 507–517 (1996)
Bisognin, E., Bisognin, V., Perla Menzala, G.: Asymptotic behaviour in time of the solutions of a coupled system of KdV equations. Funkc. Ekvacioj 40, 353–370 (1997)
Bisognin, E., Bisognin, V., Sepúlveda, M., Vera, O.: Coupled system of Korteweg-de Vries equations type in domains with moving boundaries. J. Comput. Appl. Math. 220, 290–321 (2008)
Bona, J., Ponce, G., Saut, J.C., Tom, M.M.: A model system for strong interaction between internal solitary waves. Commun. Math. Phys. Appl. Math. 143, 287–313 (1992)
Bourgain, J.: Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I Schrödinger equation. Geom. Funct. Anal. 3, 107–156 (1993)
Craig, W., Kappeler, T., Strauss, W.: Gain of regularity for equations of Korteweg-de Vries type. Ann. Inst. Henri Poincaré 2, 147–186 (1992)
Dávila, M.: Continuação única para um Sistema Acoplado de Equações do tipo Korteweg-de Vries e para as Equações de Benjamin-Bona-Mahony e de Boussinesq. Tese de Doutorado, IM-UFRJ, Brazil (1995)
Gear, J.A., Grimshaw, R.: Weak and strong interactions between internal solitary waves. Commun. Math. 70, 235–258 (1984)
Kato, K., Ogawa, T.: Analyticity and smoothing effect for the Korteweg-de Vries equation with a single point singularity. Math. Ann. 316(3), 577–608 (2000)
Kato, T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equations. Adv. Math. Suppl. Stud., Stud. Appl. Math. 8, 93–128 (1983)
Kato, T., Masuda, K.: Nonlinear evolution equations and analyticity. I. Ann. Inst. Henri Poincaré Anal. Non-linéaire 3, 455–467 (1986)
Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)
Kenig, C., Ponce, G., Vega, L.: A bilinear estimate with applications to the KdV equation. J. Am. Math. Soc. 9, 573–603 (1996)
Kenig, C., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction mapping principle. Commun. Pure Appl. Math. 46, 527–620 (1993)
Kenig, C., Ponce, G., Vega, L.: On the (generalized) Korteweg-de Vries equation. Duke Math. J. 59(3), 585–610 (1989)
Kenig, C., Ponce, G., Vega, L.: Oscillatory integrals and regularity equations. Indiana Univ. Math. J. 40, 33–69 (1991)
Linares, F., Panthee, M.: On the Cauchy problem for a coupled system of KdV equations. Commun. Pure Appl. Ann. 3(3), 417–431 (2004)
Vera, O.: Gain of regularity for a generalized coupled system of nonlinear evolution dispersive equations type. Ph.D. Thesis, UFRJ, Rio de Janeiro, Brazil (2001)
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Alves, M.S., Calsavara, B.M.R., Muñoz Rivera, J.E. et al. Analyticity and Smoothing Effect for the Coupled System of Equations of Korteweg-de Vries Type with a Single Point Singularity. Acta Appl Math 113, 75–100 (2011). https://doi.org/10.1007/s10440-010-9586-2
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DOI: https://doi.org/10.1007/s10440-010-9586-2