The Stability Of The Solutions With 2-D Random Artificial Compressible Navier-Stokes Equation

The stability theory of stochastic differential equations is developed based on the stability theory of deterministic differential equation and stochastic process theory.At the same time,the infinite dimensional dynamic system of random perturbation has been studied more and more in recent years and has been widely used in mechanics,chemistry,biology,geophysics,atmospheric Marine climatology and so on.In recent years,the study of the deterministic Navier-Stokes equations has attracted the attention of scholars.On the other hand,due to earthquake,flood,migration and instantaneous plague,the study of the Navier-Stokes equations with random perturbation has aroused more and more interest.The research contents of this paper have the following aspects.Firstly,the background and research status of stochastic differential equation theory and related knowledge and the Navier-Stokes equation under random perturbation are introduced.Secondly,the definition of generalized solutions of Navier-Stokes equations under random perturbation is given.By using the properties of continuous martingale,the It? Formula,Gronwall Lemma,and Burkholder-Davis-Gundy Inequality are used to deal with the pressure term ,and the exponential stability of the solution of random Navier-Stokes equations is discussed,and sufficient conditions for exponential stability are obtained.Thirdly,by using the generalized Gronwall extension lemma and the synthesis of both 1)()and 2)(,),the exponential stability of the solution of random Navier-Stokes equations is discussed,and the sufficient condition of the exponential stability is obtained.