Abstract
This paper presents the existence and uniqueness of solutions to an abstract nonlinear equation driven by multiplicative noise of Lévy type. This equation covers many hydrodynamical models, including 2D Navier-Stokes equations, 2D MHD equations, the 2D Magnetic Bernard problem, and several Shell models of turbulence. In the literature on this topic, besides the classical Lipschitz and linear growth conditions, other atypical assumptionsare also required on the coefficients of the stochastic perturbations. The goal of this paper is to eliminate these atypical assumptions. Our assumption on the coefficients of stochastic perturbations is new even for the Wiener cases and, in one sense, is shown to be quite sharp. A new cutting off argument and energy estimation procedure play an important role in establishing the existence and uniqueness under this assumption.
Funding Statement
Xuhui Peng was supported by the National Natural Science Foundation of China (NSFC) (No. 12071123), Scientific Research Project of Hunan Province Education Department (No. 20A329), and Construction Program of the Key Discipline in Hunan Province. Juan Yang is corresponding author and supported by NSFC (Nos. 11871010, 11871116). Jianliang Zhai was supported by NSFC (Nos. 12131019, 11971456, 11721101), the Fundamental Research Funds for the Central Universities (No. WK3470000016), and the School Start-up Fund (USTC) KY0010000036.
Acknowledgments
The authors are very grateful to Professors Ping Cao and Yong Liu for their valuable suggestions.
Citation
Xuhui Peng. Juan Yang. Jianliang Zhai. "Well-posedness of stochastic 2D hydrodynamics type systems with multiplicative Lévy noises." Electron. J. Probab. 27 1 - 31, 2022. https://doi.org/10.1214/22-EJP779
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