Random Dynamics of the Stochastic Boussinesq Equations Driven by Lévy Noises

Abstract

This paper is devoted to the investigation of random dynamics of the stochastic Boussinesq equations driven by Lévy noise. Some fundamental properties of a subordinator Lévy process and the stochastic integral with respect to a Lévy process are discussed, and then the existence, uniqueness, regularity, and the random dynamical system generated by the stochastic Boussinesq equations are established. Finally, some discussions on the global weak solution of the stochastic Boussinesq equations driven by general Lévy noise are also presented.

Strong solutions for the stochastic Navier-Stokes equations on the 2D rotating sphere with stable Lévy noise

“…Lemma 3.7 (Corollary 8.1, [14]). Assume that: p ∈ (1, 2]; X is a subordinator Lévy process from the class Sub(p); E is a separable type p Banach space; U is a separable Hilbert space; E ⊂ U ; and W = (W (t), t ≥ 0) is an U -valued Wiener process.…”

Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise

“…ω ∈ D(R, X),z(ϑ s ω)(t) =ẑ(ω)(t + s), t, s ∈ R.Proof. The proofs of the first three parts follows from closely from Theorem 4.8 and Proposition 8.4 in[9], see also Theorem 9 in[23]. For the last part, since (ϑ s ω)(r) = ω(r + s) − ω(s), r ∈ R, we havê…”

Additive noise destroys the random attractor close to bifurcation

“…The existence of pull-back random attractors has been extensively studied by many authors for several kinds of SPDEs defined on bounded domain (see, e.g. [16][17][18]). For the unbounded domain case, the situation becomes much more complicated, and we need to deal with more difficulties caused by the unboundedness of the spacial domain.…”