Local-in-time well-posedness for 2D compressible magneto-micropolar boundary layer in Sobolev spaces
Yuming Qin, Junchen Liu
TL;DR
The paper establishes local-in-time well-posedness for the 2D compressible magneto-micropolar boundary layer on the half-plane in Sobolev spaces by reformulating the problem to overcome derivative loss in the tangential direction. A stream function is used to couple the magnetic field components, and a nonlinear coordinate transform yields a Prandtl-type system in reformulated variables, enabling energy methods. A Picard iteration with uniform Sobolev estimates and a compactness argument yields a unique local solution, which is then mapped back to the original variables to prove existence for the boundary-layer equations. The approach hinges on non-degeneracy of the initial tangential magnetic field and careful control of boundary terms through symmetric matrices and energy estimates. Overall, the work extends the boundary-layer well-posedness theory to a compressible magneto-micropolar setting in Sobolev spaces without monotonicity assumptions on the tangential fields.
Abstract
In this paper, we study the two-dimensional compressible magneto-micropolar boundary layer equations on the half-plane, which are derived from 2D compressible magneto-micropolar fluid equations with the non-slip boundary condition on velocity, Dirichlet boundary condition on micro-rotational velocity and perfectly conducting boundary condition on magnetic field. Based on a nonlinear coordinate transformation proposed in \cite{LXY2019}, we first prove the local-in-time well-posedness for the compressible magneto-micropolar boundary layer system in Sobolev spaces, provided that initial tangential magnetic field is non-degenerate.