Stochastic 2-D Navier--Stokes Equation

Abstract

Abstract. In this paper we study the stochastic Navier-Stokes equation with artificial compressibility. The main results of this work are the existence and uniqueness theorem for strong solutions and the limit to incompressible flow. These results are obtained by utilizing a local monotonicity property of the sum of the Stokes operator and the nonlinearity.

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

“…Thus, this result contains the corresponding existence and uniqueness theorems and a priori bounds for 2D Navier-Stokes equations (see, e.g. [28,34]), for the Boussinesq model of the Bénard convection (see [17], [14]), and also for the GOY shell model of turbulence (see [1] and [27]). Theorem 2.4 generalizes the existence result for MHD equations given in [2] to the case of multiplicative noise and also covers new situations such as the 2D magnetic Bénard problem, the 3D Leray α-model and the Sabra shell model of turbulence.…”

“…However the hypotheses in [12] do not cover our hydrodynamical type model. We also note the stochastic Navier-Stokes equations were studied by many authors (see, e.g., [5,18,28,36] and the references therein).…”

“…Our argument mainly follows the local monotonicity idea suggested in [28,34]. However, since we deal with an abstract hydrodynamical model with a forcing term which contains a stochastic control under a minimal set of hypotheses, the argument requires substantial modifications compared to that of [34] or [27].…”

Asymptotic behavior of solutions of stochastic evolution equations for second grade fluids

“…Since we are in 2-D then it is known that under our hypotheses (I)-(III) the problem (5) has a probabilistic strong solution which is unique, see for example [9]. This implies that the process v of the above theorem is a probabilistic strong solution of the stochastic Navier-Stokes equations (5).…”

Large deviations for the stochastic shell model of turbulence

“…Menaldi and Sritharan [27] further developed this theory for the case when the sum of the linear and nonlinear operators are locally monotone. …”