Existence And Long Time Behavior Of Solutions For Fractionally Dissipative Quasi-geostrophic Equations

The quasi-geostrophic equation,which is an important model to describe geo-physical fluid dynamics,arises in the study of the evolution of potential temperature with incompressible fluid in atmospheric flow.It plays an important role in theoret-ical research,meteorology and oceanography.Therefore,this dissertation investigate the existence and long time behavior of solutions for some kings of quasi-geostrophic modelsThis dissertation is divided into six chapters.In the first chapter,we first sum-marize the development and the current research situation of theories related to quasi-geostrophic equations,and show the main contents,research methods and innovation points of this dissertation.Then we introduce some notations,and briefly recall some useful estimates and relevant preliminaries from functional analysis and stochastic analysis theoryIn the second chapter,we aim to formulate an abstract result which can be used to treat the solutions of critical and super-critical equations.In both cases we first improve the viscosity and solve the regularization of the equation by using Dan Henry's technique,then passe to the limit in the improved viscosity term to get a solution of the limit problem.While in critical case we just consider a bit higher fractional power of the viscosity term,for super-critical case we need to use a version of the"vanishing viscosity technique".The abstract result is also illustrated with the 2D quasi-geostrophic equation and the Navier-Stokes equation.Finally,we show that there exists a compact global attractor for the semiflow generated by solutions to the critical quasi-geostrophic equationIn the third chapter,a fractionally dissipative 2D quasi-geostrophic equation with an external force containing infinite delay is considered in the space Hs with s? 2-2?and ? ?(1/2,1).First,we investigate the existence and regularity of solutions by Galerkin approximation and the energy method.The continuity of solutions with respect to initial data and the uniqueness of solutions are also established.Then we prove the existence and uniqueness of a stationary solution by the Lax-Milgram theorem and the Schauder fixed point theorem.Using the classical Lyapunov method,the construction method of Lyapunov functionals and the Razumikhin technique,we analyze the local stability of stationary solutions.Particularly,the polynomial stability of stationary solutions is verified in a particular case of unbounded variable delay.Finally,we prove a new generalized integral inequality,and give the general stability of the solution for such system when the delay is a bounded measurable function and the diffusivity coefficient is time varying,including the exponential stability,the polynomial stability and the logarithmic stability.In the fourth chapter,we consider a stochastic fractionally dissipative quasi-geostrophic equation driven by a multiplicative white noise in the space HS with s?2-2? and ? ?(1/2,1),whose external forces contain some hereditary char-acteristics.In order to overcome difficulties caused by the quadratic nonlinear term,a modified version is introduced.First we investigate global martingale solutions of the modified system by the classical Faedo-Galerkin approximation,the compactness method,the Skorohod theorem and the martingale representation theorem.Then the pathwise uniqueness of the martingale solution is established.Finally we show the existence of the pathwise solution based on the pathwise uniqueness of martingale so-lutions and the Yamada-Watanabe theorem.For the critical case ?=1/2,we obtain the similar results in Hs with s>1.In the fifth chapter,the existence and uniqueness of pathwise solutions for a s-tochastic fractionally dissipative quasi-geostrophic equation driven by a multiplicative white noise are proved in Hs,where s? 2-2? and ? ?(1/2,1).Moreover,we show the exponential stability in q-th moment of ?·?Lq with q>2/(2?-1)and almost sure exponential stability in Lq for the stochastic quasi-geostrophic equation.The stabiliza-tion effect on the deterministic system under the presence of stochastic perturbations is also analyzed.Finally,we establish the connection between the deterministic sys-tem and its random perturbations by considering the limiting behavior of invariant measures for stochastic quasi-geostrophic equations with small noise intensity.In the sixth chapter,we consider a stochastic fractionally dissipative quasi-geostrophic equation with stochastic damping.First,we show that the null solution is exponential-ly stable in the sense of q--th moment of ?·?Lq,where q>2/(2?-1)and q-denotes the number strictly less than q but close to it,and from this fact we further prove that the sample paths of solutions converge to zero almost surely in Lq as time goes to infinity.Then the uniform boundedness of pathwise solutions in Hs with s? 2-2?and ? ?(1/2,1)is established,which implies the existence of non trivial invariant measures.Finally,we investigate the ergodicity of invariant measures in the case of degenerate additive noise.