Sure And Uncertainty Of The Navier-stokes Equations In A Number Of Issues

We study the initial boundary value problems for deterministic Navier-Stokes equations and nondeterministic Navier-Stokes equations with time delay. The systems come from the physical consideration.In the deterministic case, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations with a gravitational force and degenerate viscosity coefficients. We apply some new techniques to deal with the difficulties coming from such system with variable viscosity coefficients. Under certain assumptions that are imposed on the initial data, we obtain the global existence and uniqueness of the weak solution and give some uniform bounds (with respect to time) of the solution. Moreover, we obtain some stabilization rate estimates in L~∞-norm and weighted H~1-norm of the solution. Our results show that such a system is stable under small perturbations, it does not develop vacuum states or concentration states for all time, and the interface propagates with finite speed.In the nondeterministic case, due to some inertia and random, the nondeterministic Navier-Stokes equation with the nondeterministic external force with time delay are considered. The global existence of weak solution of two-dimensional and three-dimensional nondeterministic Navier-Stokes equation are obtained by using Galerkin method and compactness principle. In the two-dimensional nondeterministic case, the uniqueness and the continuity with respect to the initial data of the weak solution are established. Furthermore, we study the asymptotic behavior of solutions to the 2-dimensional Navier-Stokes equation with the nondeterministic external force with time delay as a disturbance; Some results of the mean square exponential stability and almost surely exponential stability are obtained.