The Free-Boundary Problems in Inviscid Magnetohydrodynamics with or without Surface Tension

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Date
2022-03-07
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Johns Hopkins University
Abstract
The free-boundary problems in magnetohydrodynamics (MHD) describe the motion of conducting fluids in electromagnetic fields. Such problems usually arise from the plasma confinement problems and some astrophysical phenomena, e.g., the propagation of solar wind. The thesis records the results for the local well-posedness (LWP) of the free-boundary problems in incompressible MHD with and without surface tension [52, 53, 28, 29] (joint with Xumin Gu and Chenyun Luo), compressible resistive MHD [82, 83], and compressible ideal MHD [50] (joint with Hans Lindblad). For incompressible ideal MHD, we record a comprehensive study for the case with surface tension [53, 28, 29] which are the first breakthrough in the mathematical study of this direction. The proof relies on the tangential smoothing, penalization method and a new-developed cancellation structure enjoyed by the Alinhac good unknowns. When the surface tension is neglected, we present a minimal regularity result (for LWP) in a small fluid domain [52]. Compressible ideal MHD is a hyperbolic system with characteristic boundary conditions. When the magnetic field is parallel to the surface, the loss of normal derivatives cannot be compensated due to the failure of div-curl analysis. On the one hand, we observe that such derivative loss is exactly compensated by the magnetic diffusion. Based on this, we prove the LWP and the incompressible limit for compressible resistive MHD [82, 83]. On the other hand, we adopt the anisotropic Sobolev spaces together with the “modified" Alinac good unknowns to study compressible ideal MHD system. We establish the first result [50] on the nonlinear a priori estimates without loss of regularity for the free-boundary compressible ideal MHD system, which greatly improves the existing results proved by Nash-Moser iteration.
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Keywords
Partial Differential Equations, Magnetohydrodynamics, Fluid Dynamics, Surface Tension, Inviscid Flows, Free-Boundary Problems.
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