Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type

Abstract

Abstract. In this paper we prove the existence and uniqueness of maximal strong (in PDE sense) solution to several stochastic hydrodynamical systems on unbounded and bounded domains of 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-α model with jump type perturbation. Our goal is achieved by proving general results about the existence of maximal and global solution to an abstract stochastic partial differential equations with locally Lipschitz continuous coefficients. The method of the proofs are based on some truncation and fixed point methods.

Stochastic one layer shallow water equations with Lévy noise

“…[23]. A general treatment of stochastic hydrodynamical systems with Lévy jump noise is given in [3], which overlaps with the current work without either paper being contained in the other. Both the current article and [3] show the existence of a maximal local pathwise solution.…”

Large Deviations for Stochastic Nematic Liquid Crystals Driven by Multiplicative Gaussian Noise

“…Notation. For a stopping time τ we set 1 In the sense that for all n ∈ N, τn ≤ τn+1, P-a.s. Definition 3.5.…”

“…We should notice that some of the arguments elaborated in Section 5 have been already used in [1] and [5] which respectively studied the strong solution of some stochastic hydrodynamic equations (NSEs, MHD and 3D Leray α-models) driven by Lévy noise, and the existence and uniqueness of a maximal local smooth solution to the stochastic Ericksen-Leslie system (1.1)-(1.4) on the d-dimensional torus. We are also strongly convinced that with these general results it is possible, although it has not been done in detail, to prove the existence of strong solution of several stochastic hydrodynamical models such as the NSEs, MHD equations, α-models for Navier-Stokes and related problems.…”

Some results on the penalised nematic liquid crystals driven by multiplicative noise: weak solution and maximum principle

“…Therefore, the present article is a generalization of [9] in the sense that we allow f (·) to be a general polynomial function. Note that some of the arguments from the last part of the present paper have already been used in the paper [4] which studied the strong solution of some stochastic hydrodynamic equations driven by Lévy noise.…”