Stochastic Partial Differential Equations and Turbulence

Abstract

Stochastic partial differential equations are proposed in order to model some turbulence phenomena. A particular case (the stochastic Burgers equations) is studied. Global existence of solutions is proved. Their regularity is also studied in detail. It is shown that the solutions cannot possess too high regularity.

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

“…(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 }.…”

“…In applications, the present approach allows to consider the multiplicative Gaussian noise term, represented by t 0 in the noise term is important in modelling the turbulence, see [8] and [38]. Assumptions (G.1)-(G.3) formulated in Section 2 cover the following example…”

“…[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].…”

Existence of global stochastic flow and attractors for Navier–Stokes equations

“…The paper [8] dealt exclusively with the existence of solutions for such g. Its precursors [5,6] (see also Sec 6.3 of [9]) also gave a physical motivation for considering noise of this form.…”

“…Then for each v ∈ H and s ∈ R the processu(t, ω) = ψ(t, s, v, ω) is a solution to the equation (2.1) on [s, ∞) with u(s) = v. This process satisfies (1) for all t ≥ s and all ω |u(t)| 2 + c 1 t s u(r) 2 dr ≤ |u(s)| 2 exp(α(t − s)) + c 4 (exp(α(t − s)) − 1), (5.7) (2) if α < λ 1 (2ν − β), then for all t ≥ s and all ω |u(t)| 2 ≤ |u(s)| 2 exp(−c 3 (t − s)) + c5 1 − exp(−c 3 (t − s)) , (5.8) (3) for all t ≥ s, v, v ∈ H |u (t) − u(t)| 2 + t s u (r) − u(r) 2 dr ≤ |v − v| 2 c(t − s, |u|), (4) assuming F2 and G4 p , then for all T ≥ s, all v ∈ V E sup s≤t≤T u(t) p ≤ c p (T − s, v ),(5.9)(5) assuming F2 and G4 2p , then for all T ≥ s and v ∈ V c p (T − s, v ),(5.10) …”