Abstract
We study the global well-posedness and wave-breaking phenomenon for the stochastic rotation-two-component Camassa–Holm (R2CH) system. First, we find a Hamiltonian structure of the R2CH system and use the stochastic Hamiltonian to derive the stochastic R2CH system. Then, we establish the local well-posedness of the stochastic R2CH system using a dispersion-dissipation approximation system and the regularization method. We also show a precise blow-up criterion for the stochastic R2CH system. Moreover, we prove that the global existence of the stochastic R2CH system occurs with high probability. At the end, we consider the transport noise case and establish the local well-posedness and another blow-up criterion.
Funding Statement
Yong Chen was supported by China NSF Grant Nos. 12071434, 12071435 and Zhejiang Provincial NSF of China under Grant No. LZJWY22E060002.
Hongjun Gao was supported in part by China NSF Grant No 12171084 and the fundamental Research Funds for the Central Universities No. 2242022R10013.
Acknowledgments
The authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.
Hongjun Gao is the corresponding author.
Citation
Yong Chen. Jinqiao Duan. Hongjun Gao. "Well-posedness and wave-breaking for the stochastic rotation-two-component Camassa–Holm system." Ann. Appl. Probab. 33 (4) 2734 - 2785, August 2023. https://doi.org/10.1214/22-AAP1877
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