Well-posedness of stochastic Korteweg-de Vries-Benjamin-Ono equation

Abstract

In this paper, we consider the stochastic Korteweg-de Vries-Benjamin-Ono equation with white noise. Using Fourier restriction norm method and some basic inequalities, we obtain a local existence and uniqueness result for the solution of this problem. We also get global existence of the L²(ℝ) solution.

Keywords

Korteweg-de Vries-Benjamin-Ono (KdV-BO) equation / Fourier restriction norm / [k;Z]-multiplier / cylindrical Wiener process

References

[1.] Bergh J., Löfström J. Interpolation Spaces, 1976, Berlin: Springer-Verlag.

[2.] de Bouard A., Debussche A. On the stochastic Korteweg-de Vries equation. J Funct Anal, 1998, 154: 215-251.

[3.] de Bouard A., Debussche A., Tsutsumi Y. White noise driven Korteweg-de Vries equation. J Funct Anal, 1999, 169: 532-558.

[4.] Chang H. Y., Lien C., Sukarto S., Raychaudhury S., Hill J., Tsikis E. K., Lonngren K. E. Propagation of ion-acoustic solitons in a non-quiescent plasma. Plasma Phys Control Fusion, 1986, 28: 675-681.

[5.] Guo B., Huo Z. The well-posedness of the Korteweg de Vries-Benjamin-Ono equation. J Math Anal Appl, 2004, 295: 444-458.

[6.] Herman R. L. The stochastic, damped Korteweg-de Vries equation. J Phys A, 1990, 23: 1063-1084.

[7.] Kenig C. E., Ponce G., Vega L. The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math J, 1993, 71: 1-21.

[8.] Kenig C. E., Ponce G., Vega L. A bilinear estimate with application to the Korteweg-De Vries equation. J Amer Math Soc, 1996, 9: 573-603.

[9.] Kim J. U. On the stochastic Benjamin-Ono equation. J Diff Eq, 2006, 228(2): 737-768.

[10.] Linares F. L² Global well-posedness of the initial value problem associated to the Benjamin equation. J Diff Eq, 1999, 152: 1425-1433.

[11.] Matsuno Y. Stochastic Benjamin-Ono equation and its application to the dynamics of nonlinear random waves. Phys Rev E, 1996, 52(6): 6313-6322.

[12.] Da Prato G., Zabczyk J. Stochastic Equations in infinite Dimensions, 1992, Cambridge: Cambridge University Press.

[13.] Scalerandi M., Romano A., Condat C. A. Korteweg-de Vries solitons under additive stochastic perturbations. Phys Rev E, 1998, 58: 709-712.

[14.] Tao T. Multilinear weighted convolution of L² functions, and applications to nonlinear dispersive equation. Amer J Math, 2001, 123: 839-908.