Global Well-posedness Of The Three Dimensions Incompressible Liquid Crystal Flows

In this dissertation,we study the global well-posedness of the 3D nematic liquid crystal flows.Firstly,we consider the following two kinds of simplified Ericksen-Leslie model:The first kind of the simplified Ericksen-Leslie model can be written as#12 where u=(u1(x,t),u2(x,t),u3(x,t)is the velocity field,d=(d1(x,t),d2(x,t),d3(x,t))is the macroscopic average of molecular orientation field and p represents the scalar pressure.And ? 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.The notation ?d??d represents the 3 x 3 matrix,of which the(i,j)th component can be denoted by(?)dk(i,j?3).Next,we introduce the second kind of the simplified Ericksen-Leslie model as#12 where f(d)=1/?2(1-|d|2)d,and ? is a positive constant.Since the system(0.0.1)and(0.0.2)contain the Navier-Stokes equations as a sub-system,in general,the global existences of smooth solution for the two systems above are also difficult problems.In the third chapter7 we study the global regularity of the above two equations,and by using standard energy method,together with basic in-equalities,we get two regularity criteria:(1)If the local strong solution of the system(0.0.1)satisfies ?0T(??3?Lp2p/2p-3+?u3?Lq 2q/q-3+??d?B?,?0 2)dt<?,3/2<p??,3<q??,the solution is global;(2)If the local smooth solution of the system(0.0.2)satisfies?0T??uh?Lpqdt<?,3/p+2/q?3/2,2?p??,where uh=(u1,u2),the solution is regular.Besides,we consider the following 3D generalized incompressible liquid crystal equa-tions with fractional Laplacian dissipation:where ?,? are the nonnegative diffusion indices,operator(-?)? is defined through theFourier transform:(-?)?f(?)=|?|2?f(?),and f represents the Fourier transform of f.In the fourth chapter,we consider the global well-posedness problem of the system(0.0.6).By employing commutator estimates and some basic inequalities,combining with energy method,we obtain a result:assume the initial data(u0,d0)? Hs(R3)×Hs+1(R3),s>5/2,and ?·u0=0,if ??5/4,??5/4,then there exists a unique global strong solution u,?d ? L?(0,T;Hs(R3)).