Some Problems Associated With Stochastic Partial Differential Equations In Fluid Dynamics

In this thesis,we study the incompressible stochastic Navier-Stokes equations and stochastic equations of second grade fluids on a bounded domain of R2.The Navier-Stokes equations describe the motion of Newtonian fluids.The second grade fluids equations describe the motion of a class of non-Newtonian fluids.To begin with,we investigate the stochastic Navier-Stokes equations and study a class of approximation problems,that is,we will show that the solutions of stochastic Navier-Stokes equations driven by pure jump noise can approximate to the solutions of stochastic Navier-Stokes driven by Brownian motion.We obtain the above approxima-tion results under appropriate conditions.The difficulty of proving this approximation problem lies in establishing the tightness of the solutions to the approximating equa-tions in the space of L2-valued right continuous paths with left limits.To overcome this difficulty,we first assume that the initial value,the external force and the jump coefficients take values in a more regular space,so that we derive a uniform estimate of the stronger norm of the solutions to the approximating equations.Using these esti-mates,the tightness of the solutions to the approximating equations is established.Then,through martingale characterization,it is proved that the limit of the solutions to the ap-proximating equations is the solution of the stochastic Navier-Stokes equations driven by Brownian motion.Afterwards,we approximate the initial value,the external force and the jump coefficients by finite dimensional projections,and obtain some uniform convergence of the solutions of the equations corresponding to the finite dimensional projections in the sense of probability.We can then remove the restriction put on the initial value,the external force and the jump coefficients.In the second part,we investigate the stochastic equations of second grade fluids,which consist of the following three aspects.First,we establish the existence and uniqueness of strong probabilistic solutions to the stochastic equations of second grade fluids driven by Lévy noise.The variational approach is employed here.We show that equations of second grade fluids satisfy the local monotonicity condition.However,the existing variational frameworks in the liter-ature could not cover the situation considered in this thesis.The reason is that equations of second grade fluids do not satisfy the coercivity condition or generalized coercivity condition required in the literature.In this thesis,we obtain a new orthogonality prop-erty for the nonlinear curl-term(i.e.the curl-term is orthogonal to the solution with re-spect to the inner product,here the inner product induces an equivalent norm of Sobolev space W3,2),thus establishing a uniform W3,2-norm estimate of the Galerkin approx-imations.And then the corresponding convergence of the Galerkin approximations is obtained.By taking full advantage of the local monotonicity,we proved the global existence and uniqueness of strong probabilistic solutions for the equations of second grade fluids driven by Lévy noise.Our results cover the case of Gaussian noise and also improve the existing corresponding results for martingale solutions in the literature,and our method used is simpler.Second,we investigate the dynamical system generated by the stochastic equations of second grade fluids driven by linear multiplicative Gaussian noise,and obtain three main results:1.the solutions generate a continuous random dynamical system,2.the system is Fréchet differentiable at the points with higher regularity of the phase space,3.the random dynamical system is asymptotically compact and possesses a random at-tractor,and the perturbed random attractors are upper semi-continuous when the noise intensity approaches zero.Due to the high nonlinearity of equations of second grade fluids and the appearance of the curl-term,the dissipation of the equations is rather weak and the solution operator is not smoothing or compact,so it is difficult to directly construct a compact and invariant random absorbing set.In this thesis,we use the expo-nential of Brownian motion to transform the stochastic equation of second grade fluids into a partial differential equation with random coefficients.On the one hand,we prove that there exists a random absorbing ball to the random dynamical system.On the other hand,the energy equation satisfied by the solution of the stochastic equation of second grade fluids is obtained.Using these results,we establish the asymptotic compactness of the random dynamical system and the existence of random attractors.Furthermore,by using the asymptotic compactness,we obtain the upper semi-continuity of the per-turbed random attractors.Third,we obtain the existence and uniqueness of solutions of the anticipating ini-tial value problem for the stochastic equation of second grade fluids driven by linear multiplicative Gaussian noise.The difficulty in proving this problem is that the Kol-mogorov continuity theorem fails in the infinite dimensional setting.Therefore,we take the following three steps.Step 1,a chain rule for the Malliavin derivative of Hilbert space-valued random variables is developed.Step 2,under reasonable conditions,a product formula for Skorohod indefinite integrals is established.Step 3,the Galerkin method is used to illustrate that the solution of the stochastic equation of second grade fluids with a deterministic initial value is Malliavin differentiable.Combining these three steps and using the exponential transformation of Brownian motion,it is proved that if the deterministic initial value is regarded as an infinite dimensional parameter of the solution to the stochastic equation of second grade fluids,then after replacing this parameter by an anticipating initial value,the process obtained is exactly the solution of the corresponding anticipating initial value problem.By using the exponential transfor-mation of Brownian motion and the product formula of Skorohod indefinite integrals,it is easy to obtain the uniqueness of solutions of the anticipating initial value problem for the stochastic equation of second grade fluids.The product formula of Skorohod indefinite integrals established in this thesis can be used to solve the problem with an-ticipating initial values and linear multiplicative noise for more general framework of stochastic partial differential equations.