Global Well-posedness Of The Incompressible Liquid Crystal Equations

In this thesis,we study two simplified Ericksen-Leslie model:(1)3D incompressible nematic liquid crystal equations:Here u is the fluid velocity field,d stands for the macroscopic average of orientation field,p is the scalar pressure,The tensorial notation ?d(?)?d denotes the 3×3 matrix whose(i,j)-th entry is given by ?xid·?xjd,nd then(?d(?)?d)ij=?k=13?xidk?xjdk for any i,j = 1,2,3.? is,the kinematic viscosity,? is the competition between the kinetic and potential energies,and ? is the microscopic elastic relation time for the molecular orientation field.If d=0,system(0.0.4)reduces to the classical Navier-Stokes.It remains unknown whether or not smooth solutions of the 3D incompressible Navier-Stokes equations can develop finite-time singularities,so the global existence of the smooth solution to system(0.0.4)is also an open problem,which has attracted many authors to study.Recently,Wen and Ding obtained that system(0.0.4)has the local smooth solution in[0,T],where T =T(?uo?Hs,?d0?Hs+1).A natural question is whether or not the local smooth solution can be extended beyond T.We will consider it in the third chapter and obtain a BKM's blow-up criterion in the Besov space of negative index,that isIf the local smooth solution of system(0.0.4)blows up in t = T,then(?)??d?B?,?02)dt=?,0<a<2.(2)Generalized liquid crystal equations with fractional diffusion where A is defined by the definition of Fourier transform as:?f(?)=|?|f(?)is the Fourier transform of f(x).The global existence of smooth solution to(0.0.2)in two dimensional space can be obtained only in the case of taking special value of ?,?.For example,?=0,?>1(Wang,Zhu[35])or ?>0,?= 1;?+? = 2,0<?<1(Jin,Zhu,Jin[401).If there is no diffusion for d,that is to say,?=0,and we enhance the dissipation given by the Laplacian operator for the velocity field u,we wonder whether or not the smooth solution to(0.0.2)is globally existent in this case.The question will be discussed in Rn in chapter Four.We establish the global existence of smooth solution to(0.0.2)not only in the case of ?=0,??1+n/2,but in the case of its weaker dissipation by a logarithmic function.More precisely,In the fourth chapter,we will consider the following generated liquid crystal equa-tions in n-dimensional space.Here the operators ?:=??/g(?)is defined by ?u(?):=|?|?/g(|?|)u(?),where ??1+n/2,g:R+?R+ satisfying 1?g(s)2?C0log(e + s),C0 is some absolute constant.The following result is obtained:Assume the initial data(u0,d0)?Hk(Rn)×Hk+1(Rn)for k>1+n/2,?·u0=0.The diffusion term for u is defined through the Fourier transform(?)where ??1+n/2,g:R+?R+ satisfying 1?g(s)2?C0log(e+s),C0 is some absolute constant.Then the equations(0.0.2)has a unique global smooth solutiol satisfying u,?d?L?(0,T;Hk(Rn)),?u?L2(0,T;Hk(Rn)).