Research On Well-posedness And Other Properties Of Weak Solutions To Some Equations Of Hydrodynamics

In this paper,we mainly discuss the content of two parts,the second chapter,the third chapter make up the fist part,in this part,we talk about the problem of energy dissipation and the existence of statistical solutions of the incompressible liquid crystals flow equations and incompressible MHD equations,where the energy dissipation comes from the well-known Onsager’s Conjecture,and has a close relation with the Kolmogorov’s theory of turbulence,besides,the concept of statistical solution can be used to explain the ensemble average in the conventional theory of turbulent.The second part contains the fourth chapter,the large-time dynamics of weak solutions to a class of compressible non-Newtonian fluids is considered in this part.In Chapter 2,for the simplified Ericksen-Leslie system,first,given any initial and boundary(or Cauchy)data(u0,d0)∈ H×H1(Ω,S2)(the initial data of direction field belongs to the upper sphere),we consider the energy dissipation problem.In dimensions three,we obtain a local equation of energy for weak solutions of liquid crystals,where we define a dissipation term D(u,d)=0 that stems from an eventual lack of smoothness in the solutions to represent the energy might disappear.At the same time,we consider the 2D case and obtain D(u,d)=0,which shows the existence of global solutions of liquid crystals equations in two dimensional in another way.Next,in a bounded domainΩ(?)IRn(n=2 or 3),using a maximal principle and Galerkin method,with the help of the compactness lemma which is similar to Foias and Teman’s construction of the homo-geneous solutions in Navier-Stokes equation,we establish the existence of homogeneous statistical solution which is a limit in some weak sense of the homogeneous statistical solutions concentrated on a periodic domain.In Chapter 3,we establish the longitudinal and transverse local energy balance equa-tion of distributional solutions of the incompressible three dimensional MHD equations.In particular,we find that the functions DLε(u,B)and DTε(u,B)appeared in the ener-gy balance,all converge to the defect distribution(in the sense of distributions)D(u,B)which has been defined in Gao-Tan,this is the dissipation term in local equation of energy for weak solutions of MHD.Further more,we give a simpler form of defect distribution term,which is similar to the relation in turbulence theory,called the "4/3-law".As a corollary,we give the analogous "4/5-law" holds in the local sense.In Chapter 4,we consider the large-time dynamics of weak solutions to a class of compressible fluids with nonlinear constitutive equations in a bounded domain Ω(?)R3,the global existence of such solutions has been showed by Feireisl,Liao and Malek.We study the large time behavior of such solutions after discussing the uniqueness of solutions to the stationary problem.