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Two‐dimensional magnetohydrodynamic equilibria with prescribed topology
Author(s) -
Laurence Peter,
Stredulinsky Edward W.
Publication year - 2000
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/1097-0312(200009)53:9<1177::aid-cpa5>3.0.co;2-a
Subject(s) - topology (electrical circuits) , mathematics , weak topology (polar topology) , sobolev space , magnetohydrodynamics , constraint (computer aided design) , compressibility , torus , extension topology , general topology , mathematical analysis , topological space , pure mathematics , physics , geometry , combinatorics , plasma , quantum mechanics , thermodynamics
We establish existence of weak solutions to the equilibrium equations of magnetohydrodynamics with prescribed topology. This is carried out in two settings. In the first we consider the variational problem of minimizing total energy in a torus under the assumption of axisymmetry, and prescription of mass and flux profiles. Existence of weak solutions implies that the prescription of topology is a natural constraint. Both the compressible and incompressible cases are considered. In the second setting we adapt examples of B. C. Low and R. Wolfson [13] and J. J. Aly and T. Amari [1, 2, 3] associated with Parker's explanation of the extreme heating of the solar corona and other solar phenomena. Existence of solutions with fixed topology is a first crucial step in rigorously examining the relationship between topology and the existence of current sheets. We use a decomposition introduced in [8, 9, 11] that captures much of the topology of level sets for certain classes of Sobolev functions. This decomposition is preserved under weak limits and so is useful for prescribing topological constraints. The approach is especially suited to the use of domain deformations. © 2000 John Wiley & Sons, Inc.