Studies Of Stochastic 2D Rotating Euler Flows And The Stability Of Some Stochastic PDEs

In this thesis,we concern the stochastic 2D rotating Euler flow and the stability of some stochastic partial differential equations driven by fractional Brownian mo-tion(fBm)with Hurst parameter H e(1/3,1/2)U(1/2,1).At last,the existence and uniqueness of the solution for the stochastic 3D Navier-Stokes equations with delay is proved.Firstly,we study the stochastic 2D rotating Euler equations with the L?(T2)ini-tial condition and the white noise initial condition,respectively.According to the 3D Navier-Stokes equations with rotation,/3-plane approximation,we deduce the 2D ro-tating Euler equations in vorticity form.By overcoming the difficulties generated by the rotation,we prove the existence and uniqueness of the solution for the equations driven by Brownian motion with L? initial condition.The stability of the solution for 0 tending to 0 is proved.Then we consider the existence of the solution for the 2D rotating Euler equations in vorticity with white noise initial condition.We de-duce the formula of N dimensional dynamics in(T2)Nassociated to the 2D Euler equations with rotation.We prove the global existence of the N dimensional dy-namics by constructing different auxiliary function and define the empirical measure:wtN:=1/(?)?i=1N?i?xiN(t).Then we use the Prohorov theorem and Skorokhod repre-sentation theorem to prove the existence of the solution in a new probability space.Secondly,we study the local exponential stability of stochastic lattice equations driven by Holder-continuous paths with Holder index H ?(1/3,1/2).The stability of a class evolution equations driven by Holder-continuous paths with Holder index H ?(1/2,1)is also proved.The fixed point theorem is used to get the existence and uniqueness of the path-area mild solution for the stochastic lattice equations.The semigroup S(t)generated by the unbounded linear operator A in the stochastic evo-lution equations is not Holder-continuous at zero.By assuming the regularity of ini-tial condition,the mild solution u(t)?G C?([0,T];W)is obtained.Then the local exponential stability of evolution equations driven by Holder-continuous paths with Holder index H?(1/2,1)is established.This theory can be applied to the stochastic differential equations or stochastic partial differential equations driven by fractional Brownian motion with Hurst parameter H ?(1/3,1/2)U(1/2,1).Finally,we study the 3D stochastic incompressible Navier-Stokes equations with delay in the convective term.We consider the 3D stochastic Navier-Stokes equations and obtain the existence and uniqueness of the weak solution.By using Ito formula and modified Gronwall inequality,we can get the uniform boundedness of Galerkin approximate system and obtain the global existence and uniqueness of the weak solu-tion.