zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] Aly, Astronom and Astrophys 221 pp 287– (1989)
[2] Aly, Astronom and Astrophys 227 pp 628– (1990)
[3] Aly, Astronom and Astrophys 319 pp 699– (1997)
[4] Bagby, J Functional Analysis 10 pp 259– (1972) · Zbl 0266.30024 · doi:10.1016/0022-1236(72)90025-0
[5] Crandall, Proc Amer Math Soc 78 pp 385– (1980) · doi:10.1090/S0002-9939-1980-0553381-X
[6] Gariepy, Arch Rational Mech Anal 67 pp 25– (1977) · Zbl 0389.35023 · doi:10.1007/BF00280825
[7] ; Axisymmetric MHDequilibria from Kruskal-Kulsrud to Grad. Variational and free boundary problems, 111-134. IMA Volumes in Mathematics and Its Applications, 53. Springer, New York, 1993. · doi:10.1007/978-1-4613-8357-4_8
[8] Laurence, C R Acad Sci Paris Sér I Math 324 pp 403– (1997) · Zbl 0897.46018 · doi:10.1016/S0764-4442(97)80076-6
[9] Laurence, C R Acad Sci Paris Sér I Math 328 pp 653– (1999) · Zbl 0938.46026 · doi:10.1016/S0764-4442(99)80229-8
[10] ; Examples of topology breaking in 3D MHD. In preparation.
[11] Laurence, Calc Var Partial Differential Equations 10 pp 197– (2000) · Zbl 0961.49023 · doi:10.1007/s005260050150
[12] ; Variational problems on unbounded domains with topological constraints. In preparation.
[13] Low, Astrophys J 324 pp 574– (1988) · doi:10.1086/165918
[14] Spontaneous current sheets in magnetic fields. International Series on Astronomy and Astrophysics, 1. Oxford University, New York 1994.
[15] Rosner, Geophys Astrophys Fluid Dynamics 48 pp 251– (1989) · Zbl 0688.76077 · doi:10.1080/03091928908218532
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.