Existence Of Stationary Statistical Solutions For Three Kinds Of Hydrodynamical Equations And Their Applications

In this paper,we study the dynamic behavior of three kinds of hydrodynamical equations from the concept stationary statistical solutions.Stationary statistical solution for some partial differential evolution equation is one Borel measure defined in energy space,which is invariant with respect to time and satisfies some sense of the corresponding partial differential evolution equation and energy inequality.It can be understood as a probability distribution of the velocity field.This concept is origin from the study of turbulence phenomena（some solutions of Navier-Stokes equations）,which can help to understand the statistical properties of solutions for the corresponding equations.Firstly,we study the time-dependent Stokes-Darcy equations on a two-dimensional bounded smooth region.We prove the existence of continuous viscous solutions for the system and attractors in the energy space under the framework of evolutionary systems proposed by Cheskidov et al.[24]（J.Differ.Equ.231（2006）,714-754）,and construct a series of stationary statistical solutions by time average.In the existing literature,the concept of steady-state statistical solution can be used to prove the nonexistence of abnormal dissipation of the two-dimensional damped Navier-Stokes equation.Abnormal dissipation denotes that when the viscosity coefficient tends to zero,the energy limit of the corresponding viscous fluid equation is greater than zero.This concept has been verified in experiments and numerical analysis and has been put forward as a basic hypothesis in turbulence theory.Up to now,there is no strict mathematical proof.In the rest part of this paper,we also apply the properties of stationary statistical solutions to study the abnormal dissipation problems of two kinds of fluid equations.In Chapter 4,we consider a class of two-dimensional damped generalized incom-pressible Navier-Stokes equations in the periodic domain with the dissipation coefficient∈(0,₂¹].We use the Yudovich uniqueness method to prove the existence and uniqueness of the weak solution of the equations and the corresponding enstrophy equations,thus the continuity of vorticity with respect to time and the existence of attractors in the solution class with special structure will be obtained.Then,by using the continuity of vorticities and the propositions of attractors,we prove the pre-compactness of the positive half or-bit,and the absence of abnormal dissipation of the system is proved in the framework of Constantin et al.[31]（Comm.Math.Phys.275（2007）,529-551）.Moreover,by using the regularization properties of the two-dimensional strongly damped SQG equation,we also prove that the energy anomalous dissipation of the solution does not exist under the slightly weak integrability condition of the initial value and the external force.In Chapter 5,we consider the axially symmetric solutions of a class of three-dimensional incompressible Navier-Stokes equations with linear damping in the whole space.Here it is assumed that the external force f and the initial value u₀ are axi-symmetric vectors and belong to H^s,S>₅/2,div u₀=0.If we also assume that the linear damping coefficient is related with the force f,we will also prove the absence of the anomalous dissipation of enstrophy of the corresponding symmetric solution for this equation.In particular,our method for proving pre-compactness of the positive semi-orbit is new and interesting.