Mathematical Sciences: Viscous Incompressible Magnetohydrodynamics: Analysis and Numerical Approximation

[404440]施密特在磁流体力学（或MHD）近似的假设下，粘性的、不可压缩的导电流体的运动由纳维-斯托克斯方程和麦克斯韦磁场方程耦合控制。后者一般将超越传导流体的区域，理想地扩展到整个空间。通常，内部和外部磁场都必须确定，并在分离流体与固体材料的界面或外部真空处适当匹配。大多数先前的数学工作关注的是流体区域表面磁场（或相关量）的边界条件给定而外部场可以忽略的特殊情况。一般情况下，除了基本的存在性和唯一性结果之外，我们所知甚少。即使这个问题可以被证明是解析性的，当试图用数值方法近似解时，也会出现巨大的困难。主要的缺点包括磁场的无界支持，以及它在分离具有不同磁性质的介质的界面上表现出跳跃不连续。拟议研究的目标是发展一般的分析和数值方法，以便在固定和时间相关的情况下严格处理这类日益复杂的问题。将研究该方法的数值和计算可行性，并与其他方法进行比较。然后将研究该方法在多大程度上适用于具有许多磁性不同介质和非光滑几何形状的更复杂情况。在后期阶段，将考虑一些扩展的MHD模型，包括非牛顿流体、霍尔电流等离子体和热耦合流动。另一个长期目标是对涉及自由边界的MHD问题进行分析和数值研究。磁流体动力学（或MHD）是研究导电流体与磁场宏观相互作用的理论。它在许多工程问题中具有重要的意义，如受控热核聚变的持续等离子体约束、核反应堆的液态金属冷却和金属的电磁铸造。它在地球物理学和天文学中也有应用，其中突出的例子是所谓的发电机问题，即地球磁场在其液态金属核心中的起源问题，以及磁性恒星在其自身引力场中的平衡和稳定性问题。本研究解决的困难在描述工程中典型应用的模型中最为明显，例如液态金属的磁成形或特殊材料的感应制造，在这些模型中，人们会遇到相当复杂的几何形状，其中有几种导电流体和载流固体导体在一定距离内通过通用磁场相互作用。

9404440 Schmidt Under the assumptions of the magnetohydrodynamic (or MHD) approximation, the motion of a viscous, incompressible, electrically conducting fluid is governed by the Navier-Stokes equations coupled to Maxwell's equations for the magnetic field. The latter will in general transcend the region of conducting fluid and, ideally, extend to all of space. Typically, both the interior and exterior magnetic fields must be determined and suitably matched at the interface separating the fluid from the solid material or vacuum outside. Most prior mathematical work has been concerned with special situations where boundary conditions for the magnetic field (or related quantities) are given on the surface of the fluid region and the exterior field can be disregarded. In the general case, little is known beyond basic existence and uniqueness results. Even if the problem can be shown to be analytically well-posed, formidable difficulties occur when trying to numerically approximate the solution. Major drawbacks include the unbounded support of the magnetic field and the fact that it exhibits jump discontinuities across interfaces separating media with different magnetic properties. It is the goal of the proposed research to develop general analytical and numerical methods tailored for the rigorous treatment of increasingly complex problems of this kind, in stationary as well as time-dependent situations. The numerical and computational feasibility of the approach will be investigated and compared to that of alternative procedures. It will then be studied to what extent the approach carries over to more complex situations featuring many magnetically different media and nonsmooth geometries. At later stages, some extended models of MHD will be considered, including non-Newtonian fluids, Hall current plasmas, and thermally coupled flows. Another long-term goal is the analytical and numerical investigation of MHD problems involving free boundaries. Magnet ohydrodynamics (or MHD) is the theory of the macroscopic interaction of electrically conducting fluids with a magnetic field. It is of importance in connection with many engineering problems, such as sustained plasma confinement for controlled thermonuclear fusion, liquid-metal cooling of nuclear reactors, and electromagnetic casting of metals. It also finds applications in geophysics and astronomy, where prominent examples are the so-called dynamo problem, that is, the question of the origin of the Earth's magnetic field in its liquid metal core, and the equilibrium and stability of magnetic stars in their own gravitational fields. The difficulties addressed by this research are most pronounced in models describing typical applications in engineering, such as the magnetic shaping of liquid metals or induction manufacturing of specialized materials, where one encounters fairly complicated geometries with several conducting fluids and current-carrying solid conductors interacting at a distance via the universal magnetic field.