Mathematical Sciences: Viscous Incompressible Magnetohydrodynamics: Analysis and Numerical Approximation

小行星9404440施密特 在磁流体动力学（或MHD）近似的假设下，粘性、不可压缩、导电流体的运动由耦合到磁场的麦克斯韦方程的Navier-Stokes方程控制。 后者一般将超越传导流体的区域，并且理想地延伸到整个空间。 通常，内部和外部磁场都必须在将流体与固体材料或外部真空分离的界面处确定并适当匹配。 大多数以前的数学工作一直关注的特殊情况下，边界条件的磁场（或相关量）的表面上的流体区域和外部场可以忽略不计。 在一般情况下，除了基本的存在性和唯一性结果外，我们所知甚少。 即使问题可以被证明是解析适定的，当试图用数值方法近似解时，也会遇到巨大的困难。 主要的缺点包括磁场的无限支撑以及它在分离具有不同磁特性的介质的界面上表现出跳跃不连续性的事实。 拟议研究的目标是开发通用的分析和数值方法，用于严格处理这种日益复杂的问题，在固定以及时间依赖的情况下。 数值和计算的可行性的方法将进行调查，并比较替代程序。 然后，将研究在何种程度上的方法结转到更复杂的情况下，具有许多磁性不同的介质和非光滑的几何形状。 在以后的阶段，将考虑一些扩展模型的MHD，包括非牛顿流体，霍尔电流等离子体，和热耦合流。 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（Magnetohydrodynamics，简称MHD）是研究导电流体与磁场之间宏观相互作用的理论。 它在许多工程问题中具有重要意义，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，以及金属的电磁铸造。 它也可以应用于天体物理学和天文学，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中的起源问题，以及磁星在其自身引力场中的平衡和稳定性问题。 这项研究所解决的困难在描述工程中典型应用的模型中最为突出，例如液态金属的磁性成形或特殊材料的感应制造，其中人们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通过通用磁场在一定距离内相互作用。

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.