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Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type. (English) Zbl 1326.60089
Summary: In this paper, we prove the existence and uniqueness of maximal strong (in PDE sense) solutions to several stochastic hydrodynamical systems on unbounded and bounded domains of \(\mathbb R^n\), \(n=2,3\). This maximal solution turns out to be a global one in the case of 2D stochastic hydrodynamical systems. Our framework is general in the sense that it allows us to solve the Navier-Stokes equations, MHD equations, Magnetic Bénard problems, Boussinesq model of the Bénard convection, Shell models of turbulence and the Leray-\(\alpha\) model with jump type perturbation. Our goal is achieved by proving general results about the existence of maximal and global solutions to abstract stochastic partial differential equations with locally Lipschitz continuous coefficients. The methods of the proofs are based on some truncation and fixed point methods.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60G51 Processes with independent increments; Lévy processes
60G57 Random measures
35R60 PDEs with randomness, stochastic partial differential equations
35Q35 PDEs in connection with fluid mechanics
35Q30 Navier-Stokes equations
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
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