Well-posedness for Stochastic Generalized Fractional Benjamin-Ono Equation

Abstract

This paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation. By using the Bourgain spaces and Fourier restriction method and the assumption that $u_{0}$ is $\mathcal{F}_{0}$-measurable, we prove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data $u_{0}(x,w)\in L^{2} (\Omega; H^{s}(\mathbb{R}))$ with $s\geq\frac{1}{2}-\frac{\alpha}{4}$, where $0< \alpha \leq 1.$ In particular, when $u_{0}\in L^{2}(\Omega; H^{\frac{\alpha+1}{2}}(\mathbb{R}))\cap L^{\frac{2(2+3\alpha)}{\alpha}}(\Omega; L^{2}(\mathbb{R}))$, we prove that there exists a unique global solution $u\in L^{2}(\Omega; H^{\frac{\alpha+1}{2}}(\mathbb{R}))$ with $0< \alpha \leq 1.$