The Limiting Behaviors Of Solutions Of Stochastic Leray-α Model Of Euler Equations

Euler equation is the most important basic equation in inviscid hydrodynamics.It has a wide range of applications and has a high research value.For example,it is widely accepted to consider uncertainty in mathematical models in other fields,such as physics,dynamic systems,mathematics and so on.In this paper,based on the deterministic Leray-a equation model.we add the transport noise and consider the stochastic Leray-a equation model of Euler equation.The limiting behaviors of the solutions of the stochastic Leray-a is studied under the scaling limit of three-dimensional transport noises.The main contents of this paper are as follows:First,the special noise is selected to make the noise tend to zero,and the well-posedness of the solution of the stochastic Leray-a equation model is studied.First of all,the existence of global weak solution of stochastic Leray-a models in the sense of distribution is studied by Galerkin approximation method.Secondly,the noise tends to zero by taking a special solution,and then the limiting behaviors of the solutions is explored by combining Prohorov theorem and Skorokhod theorem.The results show that the solution of the stochastic Leray-a model of the Euler equation with transport noise converges to the solutions of the deterministic Leray-a equation.This conclusion solves the problem of the limiting behaviors of the solution when the noise tends to zero proposed by Barbato et al.Second,on the stochastic Leray-a model in the first part,the convergence rate of the solution is studied.The central limit theorem of stochastic Leray-a model is obtained.In order to prove the quantitative convergence rate,we review the regularization effect of convolution,and then establish the quantitative convergence rate of the solution of stochastic Leray-a model by semigroup method.Finally,a strongly convergent central limit theorem is obtained by using the regularity property of random convolution,B-D-G inequality and Young inequality,and its explicit convergence rate is obtained.Last,the limiting behaviors of the solutions of stochastic Leray-α models is studied,and the large deviation of stochastic Leray-a models is established.In order to obtain large deviation by using Girsanov transform and weak convergence method,it is necessary to choose infinite dimensional noise.In this paper,we first obtain the stochastic Leray-α stochastic model with small noise from the special infinite dimensional noise,then review the weak convergence method,prove two key propositions,and use the weak convergence method to establish the large deviation of the solution of the stochastic Leray-α stochastic model with small noise.