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

Abstract

In this paper, we consider the stochastic Korteweg-de VriesBenjamin-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 L 2 (R) solution.

Well-posedness for stochastic Camassa–Holm equation

“…When H = L 2 (R) and H = H s (R), L 2 0 (L 2 (R); H s (R)) is simply denoted by L 0,s 2 . Stochastic partial differential equations and stochastic models of fluid dynamics have been the object of intense investigations, for instance, [4,[12][13][14][15]28,31].…”

A Stochastic Weakly Damped, Forced KdV-BO Equation

“…In this paper we are interested in the case when the forcing term is random. The well-posedness for the stochastic KdV-BO driven by the additive noise was studied in [6]. We have found no studies on the long time behavior of the stochastic KdV-BO equation.…”

“…Based on the global existence results given in [1], the attractor of the damped forced KdV-BO equation was obtained in [5]. The stochastic KdV-BO equation (i.e., = = 0 in (5)) was studied in space , with < 1/2 in [6]. By introducing some useful inequalities to deal with the irregularity caused by stochastic term, well-posedness results were obtained in [6].…”

“…The stochastic KdV-BO equation (i.e., = = 0 in (5)) was studied in space , with < 1/2 in [6]. By introducing some useful inequalities to deal with the irregularity caused by stochastic term, well-posedness results were obtained in [6]. Following the work in [6], we will construct the attractor to the Cauchy problem for the stochastic damped, forced KdV-BO equation.…”

“…is simply denoted by 0, 2 . The proof of the global well-posedness of the solution (5)-(6) is similar to [6]. Here, we only give the following global existence results.…”