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arXiv 1504.02172math.AP
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随机广义分数阶Benjamin-Ono方程的适定性

Well-posedness for Stochastic Generalized Fractional Benjamin-Ono Equation

Wei Yan, Jianhua Huang, Boling Guo

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This paper has been accepted for publication in SCIENCE CHINA Mathematics
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摘要：

本文研究随机广义Benjamin-Ono方程的Cauchy问题。利用Bourgain空间和Fourier限制方法，在$u_{0}$是$\mathcal{F}_{0}$ -可测的假设下，证明了随机广义Benjamin-Ono方程的Cauchy问题对于初始数据$u_{0}(x,w)\in L^{2} (Ω; H^{s}(\mathbb{R}))$和$s\geq\frac{1}{2}-\fracα{4}$是局部适定的，其中$0< α\leq 1.$特别地，当$u_{0}\in L^{2}(Ω; H^{\frac{α+1}{2}}(\mathbb{R}))\cap L^{\frac{2(2+3α)}α}(Ω; L^{2}(\mathbb{R}))$时，我们证明了存在唯一的整体解$u\in L^{2}(Ω; H^{\frac{α+1}{2}}(\mathbb{R}))$和$0< α\leq 1.$

英文摘要：

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 $u_{0}(x,w)\in L^{2} (Ω; H^{s}(\mathbb{R}))$ with $s\geq\frac{1}{2}-\fracα{4}$, where $0< α\leq 1.$ In particular, when $u_{0}\in L^{2}(Ω; H^{\frac{α+1}{2}}(\mathbb{R}))\cap L^{\frac{2(2+3α)}α}(Ω; L^{2}(\mathbb{R}))$, we prove that there exists a unique global solution $u\in L^{2}(Ω; H^{\frac{α+1}{2}}(\mathbb{R}))$ with $0< α\leq 1.$

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