The Study Of The Incompressible Viscoelastic Fluid Oldroyd-B Models

Firstly,this paper investigates the local well-posedness of the classical incompress-ible viscoelastic flow Oldroyd-B model (?)where u = u(t,x)is the velocity of the flow,p = p(t,x)is the scalar pressure,?>0 is the kinematic viscosity and F = F(t,x)is the deformation tensor of the fluid.In Chapter 3,by using Friedrichs' method,we obtain the local-in-time existence and uniqueness of strong solutions to the classical incompressible viscoelastic flow Oldroyd-B equations(0.0.4)in Hs space with s>n/2.When n = 2,3,we improve the local existence in H2 of Lin,Liu and Zhang[27].For the more general model,namely,the equation(0.0.4)2 with a damping term,(?) with v>0,similarly to the incompressible viscoelastic flow Oldroyd-B equations(0.0.4),the incompressible viscoelastic flow Oldroyd-B equations with a damping term(0.0.5)is local well-posedness.Further,this paper studies its global well-posedness with small initial data and the time decay estimates.In Chapter 4,we prove the global existence with small initial data.In Hm(m ? 3)space,firstly,we prove a priori estimate,then combine the a priori estimate with the inductive method to extend the local solution to be a global one.After obtaining the global existence,we further investigate the decay properties of the solution.By combining the equations of u and F,we estimate the decay rates of the solution.Using the Fourier-splitting method,we divide the frequency domain into two subsets:the inner sphere and the outer sphere.Applying the Fourier transform and the integral operator,we derive the optimal decay rate of the smooth solutions in L2 norm;(?)Similarly,for the higher order derivative,the frequency domain space is decomposed again by the Fourier-splitting method,and then the estimation of the decay rate of the j-th order derivative is obtained by means of an induction argument:(?)Finally,in Chapter 5,we consider the following incompressible Oldroyd-B model with six parameters (?)here the strain tensor ? = ?(x,t)is a 3 × 3 matrix,the parameter ? ? 0 is the reciprocal of relaxation time,? ? 0 is the viscosity coefficient,v ? 0 is the tensor diffusion coefficient,?>0 and q>0 are defined by the dynamic viscosity of fluid,retardation time and the parameter ?.Du is the symmetric part of ?u,namely Du = 1/2(?u +?uT).Q(?u,?)is a bilinear form and we choose Q(?u,?)=??-??+b(Du?+?Du),where ? is the antisymmetric part of ?u.The parameter b ?[-1,1].We prove the continuation criterion of a smooth solution:If u satisfies(?)with 3/2<p ? ?,2/q + 3/p ? 1,then the solution(u(x,t),?(x,t))can be extended to(0,T']for some T'>T.