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The Cauchy problem for the stochastic generalized Benjamin-Ono equation

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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 Itô formula and the Burkholder-Davis-Gundy (BDG) inequality as well as the stopping time technique.

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Acknowledgements

The first author was supported by Young Core Teachers Program of Henan Province (Grant No. 5201019430009). This work was supported by National Natural Science Foundation of China (Grant No. 11771449). The authors are deeply indebted to the referees for their helpful suggestions which greatly improve the original version of our paper.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

    Wei Yan

  2. College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, 410073, China

    Jianhua Huang

  3. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Boling Guo

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  1. Wei Yan
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  2. Jianhua Huang
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  3. Boling Guo
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Correspondence to Jianhua Huang.

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Yan, W., Huang, J. & Guo, B. The Cauchy problem for the stochastic generalized Benjamin-Ono equation. Sci. China Math. 64, 331–350 (2021). https://doi.org/10.1007/s11425-019-1620-y

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