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Existence and uniqueness for a coupled parabolic-elliptic model with applications to magnetic relaxation

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posted on 2023-06-09, 00:16 authored by David S McCormick, James C Robinson, Jose L Rodrigo
We prove existence, uniqueness and regularity of weak solutions of a coupled parabolic-elliptic model in 2D, and existence of weak solutions in 3D; we consider the standard equations of magnetohydrodynamics with the advective terms removed from the velocity equation. Despite the apparent simplicity of the model, the proof in 2D requires results that are at the limit of what is available, including elliptic regularity in $L^{1}$ and a strengthened form of the Ladyzhenskaya inequality () which we derive using the theory of interpolation. The model potentially has applications to the method of magnetic relaxation introduced by Moffatt (J. Fluid. Mech. 159, 359–378, 1985) to construct stationary Euler flows with non-trivial topology.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Archive for Rational Mechanics and Analysis

ISSN

0003-9527

Publisher

Springer Verlag

Issue

2

Volume

214

Page range

503-523

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Numerical Analysis and Scientific Computing Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2017-01-12

First Open Access (FOA) Date

2017-01-12

First Compliant Deposit (FCD) Date

2017-01-12

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