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.
Highlights
Let O ⊂ Rd be an open connected possibly unbounded subset with smooth boundary ∂O, where d = 2, 3
Using the compactness criterion in the space of càdlàg functions, we prove that a set K is relatively compact in Lq if the following three conditions hold (a) for all u ∈ K and all t ∈ [0, T], u(t) ∈ H and supu∈K sups∈[0,T] |u(s)|H < ∞, (b)
We use the result proved in [25] and following from the Jakubowski’s version of the Skorokhod Theorem [18] and the version of the Skorokhod Theorem due to Brzezniak and Hausenblas [6], see Appendix B. This will allow us to construct a stochastic process uwith trajectories in the space Z, a time homogeneous Poisson random measure ηand a cylindrical Wiener proces Wdefined on some filtered probability space ( ̄, F, P, F ) such that the system ( ̄, F, P, F, η, W, u) is a martingale solution of the problem (1–3)
Summary
We use the result proved in [25] and following from the Jakubowski’s version of the Skorokhod Theorem [18] and the version of the Skorokhod Theorem due to Brzezniak and Hausenblas [6], see Appendix B This will allow us to construct a stochastic process uwith trajectories in the space Z , a time homogeneous Poisson random measure ηand a cylindrical Wiener proces Wdefined on some filtered probability space ( ̄ , F , P , F ) such that the system ( ̄ , F , P , F , η, W , u) is a martingale solution of the problem (1–3). In Appendix C we recall Lemma 2.5 in [16] together with the proof
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