Well-posedness And Asymptotic Behavior Of The Solutions To An Incompressible Oldroyd-B Model

The present thesis is devoted to studying the Cauchy problem and the initial boundary problem for some incompressible Oldroyd-B type models.Oldroyd-B type models are widely used to describe the motion of viscoelastic fluids.First of all,we consider the Cauchy problem of three-dimensional incompressible Oldroyd-B model with stress diffusion.The global-in-time well-posedness and long-time behavior of solutions to the Cauchy problem with zero viscosity or zero stress diffusion are studied.Furthermore,we study the initial-boundary value problem for a generalized incompressible Oldroyd-B model on both two-dimensional and three-dimensional half-space.The vanishing diffusion limit is justified with convergence rates.Precisely,In Chapter 1,we first review the relevant results on the mathematical theory of the Oldroyd-B type models.Then,we introduce the main problems,results and challenges of the present thesis.In Chapter 2,we consider the Cauchy problem for three-dimensional incompressible Oldroyd-B model with stress diffusion.Firstly,based on the energy method,the global-in-time well-posedness is estiblished for fixed stress diffusion coefficient with vanishing viscosity coefficient(or for fixed viscosity coefficient with vanishing stress diffusion coefficient).Next,using the spectrum analysis method,under certain conditions,the optimal time-decay estimates of the solutions are established.It can be found that the decay speed of stress tensor is faster than velocity for the same order derivative.It is worth noticing that the decay estimates of the solutions are independent of the viscosity coefficient(for the case of vanishing viscosity coefficient)or independent of the stress diffusion coefficient(for the case of vanishing stress diffusion coefficient).In Chapter 3,we consider the initial-boundary value problem for incompressible Oldroyd-B model with stress diffusion in two-dimensional upper half-plane.Based on asymptotic analysis and boundary layer expansion,the vanishing stress diffusion limit is justified,and the corresponding optimal convergence rate is given.Moreover,the conclusion of this Chapter also shows that the boundary layer effect is weak in the limit process.Specifically,the boundary layer effect does not happen to the solution itself but to the normal derivative of the solution in the limit process.In Chapter 4,we consider the initial-boundary value problem for incompressible Oldroyd-B model with stress diffusion in three-dimensional upper half-space.Inspired by the results of two-dimensional problem in Chapter 3,for three-dimensional problem,we use different strategies to justify the vanishing stress diffusion limit.Specifically,we first prove that the solution of Oldroyd-B model is uniformly bounded(with respect to stress diffusion coefficient)in Sobolev space (21,(3 < < 6).Then,with the help of the previous uniform regularity result,we can justify the vanishing stress diffusion limit under less regularity and compatibility conditions than the results in Chapter 3.And the corresponding convergence rate is given.