Stochastic Navier–Stokes Equations Driven by Lévy Noise in Unbounded 3D Domains

Abstract

Martingale solutions of the stochastic Navier-Stokes equations in 2D and 3D possibly unbounded domains, driven by the Lévy noise consisting of the compensated time homogeneous Poisson random measure and the Wiener process are considered. Using the classical Faedo-Galerkin approximation and the compactness method we prove existence of a martingale solution. We prove also the compactness and tightness criteria in a certain space contained in some spaces of càdlàg functions, weakly càdlàg functions and some Fréchet spaces. Moreover, we use a version of the Skorokhod Embedding Theorem for nonmetric spaces.

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

“…To this end use the compactness and tightness criteria in the space Z, see Lemma 4.1 and Corollary 4.2 in Section 4. They are counterparts for the present abstract settting of the corresponding criteria proved in [39]. To prove the tightness of {L(u n ), n ∈ N} we use estimates (1.3) with p = 2.…”

“…The present paper is a straightforward generalization of the results of [39], where the stochastic Navier-Stokes equations are considered. Here, we construct an abstract framework which covers also other hydrodynamic-type equations, e.g.…”

“…stochastic magneto-hydrodynamic and Boussinesq equations. In comparison to [39] we consider more general Lévy noise term and additionally we prove estimates (1.2) on the solution of equation (1.1). Moreover, to construct a process u it is sufficient to use estimates (1.3) with p > 2 (instead of p > 4).…”

“…[7], [8], [15], [17], [25], [16], [38], [37], [41], [42] and [13]. The noise term of Poissonian type is considered in the papers [20], [19], [21] and [12], and more general Lévy noise in [39] and [45]. We consider these equations because of their importance in other hydrodynamic models, e.g.…”

“…To this end we need appropriate orthonormal basis in the space H. In the next section we recall a general approach used also in [13] and [39] in the case of Navier-Stokes equations.…”