The Cauchy problem for the stochastic generalized Benjamin-Ono equation

Abstract

The current paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation. By establishing the bilinear estimate, trilinear estimates in some Bourgain spaces, we prove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data u0(x, ω) ∈ L2 (Ω; Hs (ℝ)) which is $$\mathscr{F}_{0}$$ measurable with $$s \geqslant \frac{1}{2}-\frac{\alpha}{4}$$ and $$\Phi \in L_{2}^{0, s}$$ . In particular, when α = 1, we prove that it is globally well-posed for the initial data u0(x, ω) ∈ L2(Ω; H1(ℝ)) which is $$\mathscr{F}_{0}$$ measurable and $${\rm{\Phi }} \in L_2^{0,1}$$ . The key ingredients that we use in this paper are trilinear estimates, the Ito formula and the Burkholder-Davis-Gundy (BDG) inequality as well as the stopping time technique.