On The Global Existence And Uniqueness Of Solutions To The Non-Stationary Boundary Layer Systems

It is well known that the boundary layer systems play a very important role in the study and analysis of fluid dynamics. In recent years, considerable attention has been given to the problems by many mathematicians and physicists. There is a vast literature on theoretical, numerical and experimental aspects of the theory. Here we would like to refer the book [36] to the readers. In that book, Oleinik and Samokhin studied the stationary and non-stationary boundary layer systems including the Prandtl's system and the boundary layer system for pseudo-plastic fluids. Besides, they also studied the stationary boundary layer system for dilatable fluids. For the non-stationary case, the authers have proved the local existence of classical solutions to the Prandtl's system and the boundary layer system for pseudo-plastic fluids under some conditions. But the problems of the global existence of solutions to the systems have not yet been solved. For the non-stationary boundary layer system for dilatable fluids, all questions are open. In particular, those questions are listed as open problems in the appendix of [36]. In this paper, we mainly prove the global existence and uniqueness of weak solutions to the non-stationary boundary layer system for pseudo-plastic fluids and the Prandtl's system. We also prove the local existence and uniqueness of weak solutions to the non-stationary boundary layer system for dilatable fluids.The paper divides into six chapters. It centres on the following two important problems to the non-stationary boundary layer systems:1. The global existence of weak solutions to the non-stationary boundary layer system for pseudo-plastic fluids (including Prandtl's system), the local existence of weak solutions to the non-stationary boundary layer system for dilatable fluids;2. Uniqueness of weak solution to the non-stationary boundary layer systems. The main results in this paper are as follows:1. We prove the global existence of weak solutions to the non-stationary boundary layer system for pseudo-plastic fluids and Prandtl's system;2. We prove the local existence of weak solutions to the non-stationary boundary layer system for dilatable fluids;3. We prove the uniqueness of weak solution to the non-stationary boundary layer systems;4. We discuss the existence of weak solutions to the non-stationary boundary layer systems appearing in MHD.