In this talk, we study some Prandtl-type boundary layer models, including the two-dimensional MHD boundary layer equations and the Prandtl–Shercliff model. For small perturbations of a tangential background magnetic field, we establish the global-in-time existence and uniqueness of solutions to the MHD boundary layer equations in Sobolev spaces. The proof relies on a novel combination of the well-known cancellation mechanism and the concept of linearly good unknowns. We also investigate the Prandtl–Shercliff model. In the two-dimensional case, we establish global-in-time well-posedness in Sobolev spaces without imposing any structural assumptions on the initial data. Moreover, we show that solutions exhibit a global analytic regularization effect in all variables, up to the boundary and for all times. The proofs rely crucially on the intrinsic nonlocal diffusion induced by the Shercliff boundary layer.

Professor  Chunmei Su will talk about: 'Temporal high-order structure-preserving parametric finite element methods for curvature flows'

The quality of the mesh is crucial for simulating curvature flows, as standard approaches may fail due to mesh distortion. We first present a series of high-order parametric finite element methods based on the Barrett-Garcke--Nurnberg formulation for solving various types of flows involving curves and surfaces. Extensive numerical experiments demonstrate the anticipated high-order accuracy while maintaining favorable mesh quality throughout the evolution process. Secondly, for flows involving multiple geometric structures, such as surface diffusion—which reduces area while preserving volume—we propose a type of structure-preserving method that incorporates two scalar Lagrange multipliers along with two evolution equations related to area and volume, respectively. These schemes effectively preserve the geometric structure at a fully discrete level. Comprehensive numerical experiments illustrate that our methods achieve the desired temporal accuracy, while simultaneously preserving the geometric structure of the surface diffusion.
 

TBA; the speaker is visiting during term and this date can be flexible.