Well-posedness For The Solutions Of Oldroyd-B Model Of Complex Fluids

The present Ph.D. thesis is concerned with some analysis on hydrodynamic system for complex fluids. Complex fluids are non-Newtonian fluids, they can be viewed as the intermediate states between the fluids and the solids. The author considers the lower regularity solutions of the Oldroyd-B system of incompressible viscoelastic fluids and its global existence. This paper is divided into four parts.In Chapter 1, we introduce the background of complex fluids, the Oldroyd-B model of incompressible viscoelastic fluids, some recent results and our main works.In Chapter 2, the author considers a basic incompressible viscoelastic system which is the Oldroyd-B model with deformation tensor, it involves a system of Navier-Stokes equation coupled with nonlinear transport equation. The system is used to model materials whose elastic part of the stress tensor satisfies the Hookean law. Follow-ing [41] by Kato, we rewrite the original system into an integral equation. By means of the contraction mapping principle in the weighted Holder space we get the local well-posedness of this system provided that the initial deformation tensor is Holder continuous and the initial velocity is Lp integrable, p>d, where d is the space dimen-sion.In Chapter 3, we consider Cauchy problem to the Oldroyd-B model with stress tensorτ, and we get a Beale-Kato-Majda liked criterion which says that if the smooth solution (u,τ) to this system blows up at some finite time T*, it is necessary that∫0T*(?)u(t)(?)L∞dt=∞. Furthermore, the global existence of smooth solution to this system is given with small initial data.In Chapter 4, we study the exterior problem of the Oldroyd-B model with stress tensor, and get the global solution with small initial data.