Well-posedness for Stochastic Generalized Fractional Benjamin-Ono Equation

TL;DR

This paper establishes local and global well-posedness results for the stochastic generalized Benjamin-Ono equation using Bourgain spaces and Fourier restriction methods, under specific initial data conditions.

Contribution

It proves the well-posedness of the stochastic generalized Benjamin-Ono equation for initial data in certain Sobolev spaces, extending previous deterministic results to the stochastic setting.

Findings

Local well-posedness for initial data in $L^{2}( abla; H^{s}( ))$ with $s \\geq \\frac{1}{2}-\frac{\alpha}{4}$.

Existence of a unique global solution under additional integrability conditions.

Application of Bourgain spaces and Fourier restriction methods to stochastic PDEs.

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 $0< \alpha \leq 1.$