Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics
This book is devoted to a study of dynamics of non-Newtonian fluids with complicated rheology, in particular to visco-elastic incompressible fluids with implicit constitutive relations. The first part of the book contains preliminary material: physical background of the models (chapter 1), the necessary function spaces (chapter 2), abstract results concerning evolutionary equations in Banach spaces (chapter 3) and, finally, the theory of dynamical systems and attractors (chapter 4). Most of the presentation is self-contained and can be of independent interest, as is, for example, section 2.2 (vector-valued functions and distributions) and section 4.2 (devoted to the so-called trajectory attractors). The rest of the book is devoted to the analysis of the described models; most of the results are an extension in the spirit of the results that are known to hold for the much simpler Navier-Stokes model. Chapter 5 looks at local existence results of strong solutions, while global existence of weak (possibly non-unique) solutions and their large-time dynamics is studied in chapter 6. The analysis is completed with conditional uniqueness theorems and the existence results for the stationary case.