Global Existence And Incompressible Limit For Systems Of Viscoelastic Fluids

The present Ph.D. dissertation is concerned with some analysis on hydrodynamic systems for complex fluids. The author proves the global existence of classical solutions for two fundamental systems of viscoelastic fluids and its incompressible limits. Moreover, two rotation-strain models for viscoelastic fluids are established, and the small strain solutions are proven to exist globally in time after several underlying essential physical backgrounds are explored.First of all, the author considers a basic incompressible viscoelastic system which involves coupling between a momentum equation with dissipation mechanism and a transport equation which is not dissipative. The model is used to model materials whose stress tensor have some kind of symmetric propositions such that a strain energy function exists. The linearized equations for this system have no dissipation and scaling invariance, which lead to the main difficulties in dealing with this model. More precisely, the traditional energy methods [40] and the so-called generalized energy methods [13, 41, 42, 45, 76, 77, 78] may not be applied here. However, Lin et al. [59] recently proved the well-posedness of a viscoelastic system in two space dimensional case, and their methods can successfully be applied to deal with the difficulties coming from the partial dissipation of the system. Motivated by their ideas, the author proves the global well-posedness for general incompressible viscoelastic system, both in two dimensional and in three dimensional cases, after finding out a plenty of underlying physical backgrounds. The details can be found in chapter 2.In chapter 3, the author proves global existence of classical small strain solutions for two dimensional incompressible viscoelasticity. In fact, the rotation part of deformation tensor may not be small even for small displacements [36]. Thus, it is still of great interest to study some kind of small strain solutions. By decomposing the deformation tensorinto a strain part and a rotation part, the author establishes the rotation-strain model for the two-dimensional incompressible viscoelastic fluids, and proves that classical solutions exist globally in time provided the initial velocity and strain matrix are sufficiently small, without assumptions on the amplitude of its rotation part. The proof relies on that the dissipation in weak sense for the strain matrix and the rotational angle are found out. The results obtained there show that the smallness of the gradient of rotation angle will be enough to lead to global existence for classical solutions of two-dimensional rotation-strain model.In chapter 4, the author makes some essential improvement over the results in chapter 3. More precisely, To explain the differencce roughly, the deformation tensor is decomposed into an orthogonal part and a positive definite non-symmetric stretching part in the model in chapter 2. While the one constructed here, we still call it strain-rotation model, involves the decomposition of the deformation tensor into an orthogonal part and a positive definite symmetric stretching part, which coincides with physical origins, Fritz John's ideas [36] and others' expectations. Classical solutions are proved to exist globally in time if the initial strain and velocity are small enough. From the proof, we can also get better physical understanding of general viscoelastic systems, of the special coupling between the microscopic velocity and the internal elastic variables, of the incompressibility and of the small strain expansion in applications.In chapter 5, the author revisits the above incompressible viscoelastic model, and also study the classical incompressible viscoelastic model of Oldroyd type. The classical solutions to these two models are proven to exist golbally in time and can be viewed as the incompressible limit of solutions to the corresponding compressible systems as the Mach number tends to 0. In numerical simulations and applications, the limiting incompressible systems are often used as an approximation of the compressible ones when the Mach number is small. Thus, the results presented here are very important not only in mathematics, but also for engineering, numerical simulations and other practical applications.