Existence of a martingale solution of the stochastic Navier–Stokes equations in unbounded 2D and 3D domains

Abstract

Stochastic Navier-Stokes equations in 2D and 3D possibly unbounded domains driven by a multiplicative Gaussian noise are considered. The noise term depends on the unknown velocity and its spatial derivatives. The existence of a martingale solution is proved. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. Moreover, some compactness and tightness criteria in nonmetric spaces are proved. Compactness results are based on a certain generalization of the classical Dubinsky Theorem.

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

“…Let us denoteẽ i := e i e i U , i ∈ N. The following lemma is a straightforward counterpart of Lemma 2.4 in [13] corresponding to our abstract setting. (b) For every n ∈ N and u ∈ U 8) i.e., the restriction of P n to U is the (·, ·) U -projection onto the subspace span{e 1 , ..., e n }.…”

“…[(b i (x) · ∇)u(t, x) + c i (x)u(x)]dβ i (t), where (β) ii∈N are independent real-valued standard Wiener processes, see Section 8 in [13].…”

“…[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].…”

Weak solutions of the Stochastic Landau–Lifshitz–Gilbert Equations with nonzero anisotrophy energy

“…Our method of using the tightness criteria, the Skorokhod Theorem and the construction of the Wiener process is related but different from those applied to related problems in [11] and [12].…”

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

“…Lévy randomness requires different techniques from the ones used for Brownian motion and are less amenable to mathematical analysis. We refer to [9], [11], [20], [28] and [36] that deal with stochastic hydrodynamical systems driven by Lévy type noise. Most of these articles are about the existence of solution which are weak in the PDEs sense.…”