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 $s\ge \frac{1}{2}-\frac{\alpha }{4}$, where $0<\alpha \le 1.$ 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 $0<\alpha \le 1.$

This paper has been accepted for publication in SCIENCE CHINA Mathematics

References in corpus (2)

• A para-differential renormalization technique for nonlinear dispersive equations
• Nonlinear wave interactions for the Benjamin-Ono equation