Mathematical Sciences: RUI Problems in Magnetohydrostatic Equlilbrium Arising in the Study of the Solar Corona

9406573斯特里杜林斯基太阳天体物理学中一个突出的悬而未决的问题是太阳日冕中存在着极高的温度，约为一百万开尔文，这几代人一直困扰着天体物理学家。这个项目的主要目的是对E.N.Parker提出的一个理论进行仔细的数学分析，该理论声称日冕加热主要是由于与太阳耀斑相关的太阳磁场(电流片的形成)中某些剧烈扰动的形成。由于与太阳黑子和太阳耀斑的关系，日冕加热问题与地球气候变化和电磁干扰等实际问题直接相关。分析的主要焦点将是研究两端固定在太阳表面或光球层的管状磁力线环路。提出了当这种管子的末端扭转量超过某一临界量时，就会形成电流片。为了理解Parker日冕加热模型中的电流片形成现象，我们将在开通量管几何中研究理想三维磁流体力学方程。解的存在将取决于从管的一端到另一端的磁力线的扭转的规定。人们猜想，光滑解将存在到一定的临界扭度，此时将形成电流片，即磁场(电流)的旋度将成为奇性度量，奇性部分被一组有限的二维度量所支撑。扭度从亚临界值到临界值的转变被推测为对应于与场线结构的拓扑相联系的几何类型的索博列夫不等式中的临界指数。主要重点放在理想磁流体的恒压或无力情况下。解的存在性将通过使用迭代方案来研究，该迭代方案避免了许多变分方法所固有的紧致性的不足。

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.