On the Strong Solution for the 3D Stochastic Leray-Alpha Model

Abstract

We prove the existence and uniqueness of strong solution to the stochastic Leray-α equations under appropriate conditions on the data. This is achieved by means of the Galerkin approximation scheme. We also study the asymptotic behaviour of the strong solution as alpha goes to zero. We show that a sequence of strong solutions converges in appropriate topologies to weak solutions of the 3D stochastic Navier-Stokes equations.

Weak Solutions of a Stochastic Model for Two-Dimensional Second Grade Fluids

“…Since then, stochastic partial differential equations and stochastic models of fluid dynamics have been the object of intense investigations which have generated several important results. We refer, for instance, to [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] . Similar investigations for Non-Newtonian fluids have almost not been undertaken except in very few work; we refer, for instance, to [23][24][25] for some computational studies of stochastic models of polymeric fluids.…”

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

“…Then for r = 1, 2, for any t ≥ 0 there exists constantC > 0 such that the local solution (u, τ n ) to (2.11) satisfies 19) and 20) for any n ≥ 1.…”

“…It is worth emphasizing that the presence of the stochastic term (noise) in these models often leads to qualitatively new types of behavior for the processes. Since the pioneering work of Bensoussan and Temam [4], there has been an extensive literature on stochastic Navier-Stokes equations with Wiener noise and related equations, we refer to [1], [2], [5], [6], [16], [19], [24], [42] amongst other.…”

Strong solution for a stochastic model of two-dimensional second grade fluids: Existence, uniqueness and asymptotic behavior

“…We follow some arguments used in [8] and [14] (see also [22])). From now on we set N := N µ , F (x, t) := F (x), and G(x, t) := G(x) for any x ∈ V and t ∈ [0, T ].…”

“…Since then stochastic partial differential equations and stochastic models of Newtonian fluid dynamics have been the object of intense investigations which have generated several important results. We refer for instance to [3], [2], [10], [11], [20], [21], [22], [39], [33], [46], [44], [45]. Similar investigations for stochastic models of Non-Newtonian fluids have almost not been undertaken except in very few works.…”