On the Rate of Convergence of the 2-D Stochastic Leray- $$\alpha $$ α Model to the 2-D Stochastic Navier–Stokes Equations with Multiplicative Noise

Abstract

In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray-α model to the solution of the 2-D stochastic Navier-Stokes equations. We are mainly interested in the rate, as α → 0, of the following error functionwhere u α and u are the solution of stochastic Leray-α model and the stochastic NavierStokes equations, respectively. We show that when properly localized the error function ε α converges in mean square as α → 0 and the convergence is of order O(α). We also prove that ε α converges in probability to zero with order at most O(α).Keywords Navier-Stokes equations · Leray-α model · Rate of convergence in mean square · Rate of convergence in probability · Turbulence models · Navier-Stokes-α Mathematics Subject Classification 60H15 · 76D05 · 60H35 B Hakima Bessaih

On the Convergence of the Uniform Attractor for the 2D Leray-α Model

“…2 (R; ), then the process { 0 ( , )} corresponding to (17) with external force 0 ( , ) has the uniform ( ℎ ∈ R) attractor A 0 that coincides with the uniform ( . .…”

“…In [16], analogous results were proven for the 3D Navier-Stokes-model. In [17], the authors studied the convergence of the solution of the 2D stochastic Leraymodel to the solution of the stochastic 2D Navier-Stokes equations as approaches 0. In particular, they proved the convergence in probability with the rate of convergence at most ( ).…”

Stochastic inviscid Leray-$α$ model with transport noise: convergence rates and CLT

“…Chueshov and Millet [12] studied an abstract stochastic evolution equation of the form du + (Au + B(u, u) + R(u)) dt = √ ε σ(t, u) dW t in a Hilbert space; the conditions on the linear operator A, the bilinear mapping B and the operator R cover many 2D fluid dynamical models, as well as the 3D Leray-α model and some shell models of turbulence. We refer to [4,17] for results on the well-posedness, irreducibility and exponential mixing of stochastic 3D Leray-α model driven by pure jump noise, to [5] for α-approximation of stochastic Leray-α model to the stochastic Navier-Stokes equations, to [33] for asymptotic log-Harnack inequality and ergodicity of 3D Leray-α model with degenerate type noise. We also mention that Barbato et al [1] considered the stochastic inviscid Leray-α model, and proved that suitable transport noise restores uniqueness in law of weak solutions.…”

Stochastic 3D Leray-$α$ Model with Fractional Dissipation

“…The well-posedness and irreducibility of 3D Leray-α model driven by Lévy noise have been studied in [6,23]. The α-approximation of stochastic Leray-α model to the stochastic Navier-Stokes equations was established in [8,15]. In addition, when the viscosity constant ν = 0, Barbato, Bessaih and Ferrario in [3] studied the 3D stochastic inviscid Leray-α model.…”