Existence and uniqueness of weak solution to a three-dimensional stochastic modified-Leray-alpha model of fluid turbulence

Ridha Selmi,Rim Nasfi

doi:10.15559/21-vmsta175

Abstract

In this paper, we study the stochastic three-dimensional modified Leray-alpha model arising from the turbulent flows of fluids. We prove the existence of the probabilistic weak solution under the non-Lipschitz condition for the nonlinear forcing terms. We also discuss its uniqueness.

Highlights

In this paper, we focus on the study of the probabilistic weak solution to the following three-dimensional modified Leray-alpha model in the periodic box T = [0, 2πL]3, www.vmsta.orgR
We prove the existence of the probabilistic weak solution under the non-Lipschitz condition for the nonlinear forcing terms
Our aim in this paper is to show that the deterministic model in the case of a periodic domain introduced in [12] is reasonable, in the sense that, when some stochastic terms are present in the model, we can suggest a stochastic version with an accurate mathematical setting, yielding the existence and uniqueness of weak solutions to the problem

Summary

Introduction

In [5], the authors proved the existence and uniqueness of the variational solution to the three-dimensional stochastic Navier-Stokes-alpha (NS-α) model equations in a bounded domain, with Lipschitz assumptions on the nonlinear forcing terms. The authors of [11] obtained the existence of a probabilistic weak solution for the stochastic version of the three-dimensional Bardina model arising from the turbulent flows of fluids with non-Lipschitz conditions. To the best of our knowledge, this paper is the first work dealing with the existence and uniqueness of solution to a three-dimensional stochastic modified Leray-alpha subgrid scale model of fluid turbulence. To prove the existence of a weak solution in a Existence and uniqueness of weak solution to 3D stochastic modified-Leray-alpha model periodic domain, in Theorem 2.2, we put the system in an abstract form. The convergence results of the unique weak solution of the three-dimensional modified Leray-alpha model as the regularising parameter alpha vanishes was addressed in [22]

Functional setting and main results

Main results