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Two-dimensional magnetohydrodynamic equilibria with prescribed topology. (English) Zbl 1072.76689
Summary: 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 and J. J. Aly and T. Amari [Astron. Astrophys. 319, No. 3, 699–719 (1997; Zbl 1043.85002)] 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 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.

MSC:
76W05 Magnetohydrodynamics and electrohydrodynamics
35Q35 PDEs in connection with fluid mechanics
76M30 Variational methods applied to problems in fluid mechanics
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
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