Well-posedness Theory Of The Boundary Layer Models

The Prandtl system describes the first order approximation of the velocity field near the boundary in the zero viscosity limit of the incompressible Navier-Stokes equations.The system is the foundation of the boundary layer theory.The well-posedness theories and the justification of the Prandtl equation remain as the challenging problems in the mathematical theory of fluid mechanics.There are three difficulties of the classical Prandtl equations:due to the lack of horizontal diffusion,there exists a loss of x-derivative in the convection terms;the Poincar?e inequality does not hold for the unboundedness of the physical domain;the higher-order boundary conditions at y=0 are needed for applying the integration by parts in the y-variable,but it is a standard step to tackle it with the operator ?_t-?_y².Our main aim in this papper is to study the well-posedness theory of the 3D Prandtl system and its related models.Firstly,we mainly study the non-stationary Prandtl boundary layer of the three-dimensional axisymmetric incompressible fluid.Under Oleinik's monotonicity assumption,and we need to use axisymmetric structure u^?=0 and a new method to obtain local well-posedness.The novelty of our paper is to develop an H^s control by considering a new H^s-norm that can avoid the regularity loss created by the vertical velocity.We add the weight?1+y?for each y-derivative to our H^s energy to tackle the second difficulty.For the third difficulty we derive a reconstruction argument for higher-order boundary conditions.Next,we prove the almost global existence of classical solutions for the 3D Prandtl system with the analytic initial data in tangential variable which lie within?of a stable shear flow.Using anisotropic Littlewood-Paley energy estimates in tangentially analytic norms and introducing new linearly-good unknowns,we prove the three dimensional Prandtl system has a unique analytic solution with the life-span of which is greater than exp(?^-1/log(?^-1)).Furthermore,we consider the influence of electro-magnetic field on the boundary layer of the fluid,and obtain the almost global existence and uniqueness of the small solution for the 2D MHD boundary layer system with small initial data which are analytic in the x variable.If the initial data lie within?of a stable shear flow,then we can extend the lifespan at least up to time T_??exp(?^-1/ln(?^-1)).Unlike the classical fluids,micropolar fluids have an effect of microrotational velocity,in addition to the ordinary velocity of fluids,due to intrisic rigid rotation of material elements.Finally,we consider the boundary layer problem for the equations of motion of magnetic micropolar fluid under magnetic field.For the two-dimensional incompressible magnetic micropolar boundary layer,using a suitable change of variables and performing analytic energy estimates in the tangential variable,we obtain the local well-posedness for the 2D magneto-micropolar boundary layer system when the initial data are analytic in the x variable.