Some Topics about Magnetohydrodynamic Nonlinear Partial Differential Equations with Free Boundaries

Abstract

In this thesis, we study several free boundary problems of the incompressible magnetohydrodynamic (MHD) partial differential equations.

In Chapter 1, we give a brief introduction to the physical background of MHD and the outlines of the thesis. Meanwhile, we introduce the previous work of others to review the development of analysis on splash singularity, free-boundary problems and the application of MHD in other fields.

In Chapter 2, the existence of finite-time splash singularity is proved for the free boundary problem of the viscous incompressible magnetohydrodynamic (MHD) equations in R³, based on a construction of a sequence of initial data alongside delicate estimates of the solutions. The result and analysis in this paper generalize those by Coutand and Shkoller in [Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2019][20] from the viscous surface waves to the viscous conducting fluids with magnetic effects for which non-trivial magnetic fields (denoted by H in this thesis) may present on the free boundary. We mainly study three types of viscous incompressible MHD. The first type is the non-resistive MHD. The second type is the resistive MHD with perfectly conducting boundary conditions (H ·n = 0 and curl H × n = 0, where n is the exterior unit normal vector to the free boundary). The third type is also the MHD with resistance, with the Dirichlet boundary conditions for magnetic field (H = 0).

In Chapter 3, we study an inviscid resistive incompressible MHD equations in 2-dimensional space and prove the local well-posedness of the free-boundary problem with surface tension. We require the boundary condition of magnetic field to be perfectly conducting. To prove the existence, we construct an approximate problem in which there is an artificial viscosity on the boundary indexed by κ> 0. As  κ→0, we prove the solutions to the approximate problem will converge to that of the original MHD equations. To complete the proof of the local well-posedness, we also establish a uniform a priori estimates with respect to κ.

Date of Award	25 Aug 2025
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Tao LUO (Supervisor)