Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—Abstract framework and applications

Abstract

The existence of martingale solutions of the hydrodynamic-type equations in 3D possibly unbounded domains is proved. The construction of the solution is based on the Faedo-Galerkin approximation. To overcome the difficulty related to the lack of the compactness of Sobolev embeddings in the case of unbounded domain we use certain Fréchet space. We use also compactness and tightness criteria in some nonmetrizable spaces and a version of the Skorokhod Theorem in non-metric spaces. The general framework is applied to the stochastic Navier-Stokes, magneto-hydrodynamic (MHD) and the Boussinesq equations.

On a Stochastic 2D Simplified Liquid Crystal Model Driven by Jump Noise

“…There is a large amount of literature on the existence and uniqueness solutions for stochastic partial differential equations driven by jump-type noises. We refer the reader to [3,11,12,13,43,44,45,46,47,59,61]. However, the existing results in the literature do not cover the situation considered in this paper.…”

Existence and large time behavior for a stochastic model of modified magnetohydrodynamic equations

“…In addition to the papers we cited earlier, we would like also to mention the recent article [45] in which the spatial domain is allowed to be unbounded, and the coefficients of the noise depend multiplicatively on both the velocity field and its spatial derivatives. The key ingredients of the proof of existence of martingale solutions in [45] (see also [11,44]) are Galerkin method, the use of Fréchet space, tightness criteria in nonmetrizable spaces and a version of the Skorokhod theorem in non-metric spaces. The framework of [45] is very general as it allows to treat the stochastic Navier-Stokes, magnetohydrodynamic (MHD) and the Boussinesq equations driven by the sum of Wiener and compensated Poisson random measure.…”

“…The key ingredients of the proof of existence of martingale solutions in [45] (see also [11,44]) are Galerkin method, the use of Fréchet space, tightness criteria in nonmetrizable spaces and a version of the Skorokhod theorem in non-metric spaces. The framework of [45] is very general as it allows to treat the stochastic Navier-Stokes, magnetohydrodynamic (MHD) and the Boussinesq equations driven by the sum of Wiener and compensated Poisson random measure. However, the problem we treat here does not fall into the general framework of [45] or previous work about stochastic Navier-Stokes and MHD equations.…”

“…The framework of [45] is very general as it allows to treat the stochastic Navier-Stokes, magnetohydrodynamic (MHD) and the Boussinesq equations driven by the sum of Wiener and compensated Poisson random measure. However, the problem we treat here does not fall into the general framework of [45] or previous work about stochastic Navier-Stokes and MHD equations. The main reason is the presence of the additional nonlinear term of p-structure, which makes the mathematical analysis of (1) difficult and subtle.…”

Stochastic Control of Tidal Dynamics Equation with Lévy Noise

“…A set K ⊂ Z is τ -relatively compact if the following three conditions hold:(a) ∀u ∈ K and all t ∈ [0, T ], u(t) ∈ L 2 (O) and sup u∈K sup s∈[0,T ] u(s) L 2 < ∞, ds < ∞, i.e. K is bounded in L 2 (0, T ; H 1 0 (O)), (c) lim δ→0 sup u∈K w [0,T ],H −1 (O) (u, δ) = 0.For proof see Lemma 3.3 in[11], Lemma 4.1 in[36], Theorem 2 of[37], Lemma 2.7 in[35] …”