Multiscale models for viscoelastic fluids.

Summary This work focuses on the mathematical analysis of multi-scale models for the simulation of polymeric fluids. These models couple, at the microscopic level, a molecular description of the evolution of polymer chains (in the form of a stochastic differential equation) with, at the macroscopic level, the conservation of mass and momentum equations for the solvent (in the form of partial differential equations). Chapter 1 introduces the models and gives the main results obtained. In chapters 2, 4, 5 and 7 we show in which sense the equations are well posed for various polymer models, considering either homogeneous flows or plane sheared flows. In chapters 2, 3, 6 and 7, we analyze and prove the convergence of numerical methods for these models. Finally, chapter 8 deals with the long time behavior of the system. A second part of this paper consists of three chapters dealing with work in magnetohydrodynamics (MHD), in collaboration with industry. Chapter 9 is an introduction to the problem and to the numerical methods used. Chapter 10 describes a new test case in MHD. Finally, chapter 11 gives an analysis of the stability of the numerical scheme used.