MHD Boundary Layer Theories and Beyond
DescriptionAs the foundation of boundary layer theories, Prandtl equation was derived in 1904 from the incompressible Navier-Stokes equation with no-slip boundary condition to capture the behavior of fluid motion near the boundary when viscosity vanishes. Precisely, the Prandtl equation describes the boundary layer where the majority of the drag experienced by the solid body in a ‘simplified’ form of system derived from the incompressible Navier-Stokes equations for showing the balance between the convection and the viscosity, while outside this layer, the viscosity can be neglected as it has no significant effect on the fluid. Even though there are fruitful mathematical theories developed since the seminal works by Oleinik in 1960s, most of the well-posedness theories are limited to the two space dimensions (2D) under Oleinik’s monotonicity condition except the classical work by Sammartino-Caflisch in 1998 in the framework of analytic functions and some recent work in Gevrey function spaces.In addition to its early application in aerodynamics and later in various areas in fluid dynamics and engineering, Prandtl equation can be viewed as a typical example of partial differential equations with rich structure that includes mixed type and degeneracy in dissipation. Hence, it provides many challenging mathematical problems and many of them remain unsolved after more than one hundred years from its derivation. On the other hand, as a fundamental system for the electrically conducting fluid, MHD system has the Prandtl type system near the boundary when the Reynolds and magnetic Reynolds numbers are of the same order with the Hartmann number beingfinite. With the help of the stablilizing effect of the tangential magnetic field, we recently justify the Prandtl ansatz in Sobolev space in the 2D time dependent setting.With the understanding on the MHD boundary layer behavior, this project aims to explore its relation to other physical models, such as Navier-Stokes-Korteweg system with the capillary effect and the viscoelasticity system with the internal elasticity structure. In fact, as shown later, there is an intrinsic relation between the 2D incompressible MHD system without magnetic resistivity and the incompressible Navier-Stokes-Korteweg system, and the former system also shares some similar stability mechanismas the viscoelasticity model. Hence, we believe with the understanding on the 2D MHD boundary layer, we can initiate a new approach to study some other fluid models, such as the Navier-Stokes-Kortewag and viscoelasticity with physical boundary conditionsso that we can further study the corresponding vanishing viscosity limits.
|Effective start/end date||1/01/21 → 15/11/22|