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| In this talks, the well-posedness and dynamics of the Cauchy problem for the stochastic generalized Benjamin-Ono equation are presented. The Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data $u_{0}(x,w)\in L^{2} (\Omega; H^{s}(R))$ with $s\geq\frac{1}{2}-\frac{\alpha}{4}$, where $0< \alpha \leq 1.$ In particular, when $u_{0}\in L^{2}(\Omega; H^{\frac{\alpha+1}{2}}(R))\cap L^{\frac{2(2+3\alpha)}{\alpha}}(\Omega; L^{2}(R))$, the global well-posedness of the solution $u\in L^{2}(\Omega; H^{\frac{\alpha+1}{2}}(R))$ with $0< \alpha \leq 1$\ is also established. Our main results shows that the well-posedness depends on the order of fractional operator, the norm of Hilbert-Schmidt operator, and regularity of the initial value. Finally, the random attractor of random dynamical systems generated by the global solution is also obtained. |
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