9406573 Stredulinsky One of the outstanding open problems in solar astrophysics is the existence of enormously high temperatures in the sun's corona, on the order of a million degrees Kelvin, which have baffled astrophysicists for generations. The main object of this project is to give a careful mathematical analysis of a theory due to E.N. Parker which claims that coronal heating is primarily due to the formation of certain violent disruptions in the suns magnetic field (the formation of current sheets) associated with solar flares. Due to the relationship with sun spots and solar flares the issue of coronal heating is directly tied to the practical issues of variations in the earth's climate and electromagnetic interference. The main focus of the analysis will be the study of tube like loops of magnetic field lines with ends anchored in the suns surface or photosphere. It is proposed that current sheets form when the ends of such a tube are twisted more than a certain critical amount. In attempting to understand the phenomenon of current sheet formation in Parker's model of coronal heating, the equations of ideal 3D magnetohydrodynamics will studied in an open flux tube geometry. Existence of solutions will be considered subject to prescription of the twist in the magnetic field lines from one end of the tube to the other. It is conjectured that smooth solutions will exist up to a certain critical twist, at which point current sheets will form i.e. the curl of the magnetic field (the current) will become a singular measure, the singular part supported on a set of finite two dimensional measure. The transition from subcritical to critical values of the twist is conjectured to correspond to a critical exponent in a geometric type of Sobolev inequality linked to the topology of the field line structure. The main emphasis will be placed on the constant pressure or force free case of ideal MHD. Existence of solutions will be studied through the use of an iteration scheme which avoids the lack on compactness inherent in many variational approaches to the problem.
