Mathematical Sciences: Viscous Incompressible Magnetohydrodynamics: Analysis and Numerical Approximation

施密特磁流体力学（或MHD）是研究导电流体与磁场宏观相互作用的理论。在粘性不可压缩情况下，磁流体流动由磁场的Navier-Stokes方程和前麦克斯韦方程组控制。后者一般将超越传导流体的区域，理想地扩展到整个空间。正是流体与外界的电磁相互作用这一特征，引起了数学分析和数值近似的挑战性问题。在早期的工作中，作者开发了一种新的粘性不可压缩MHD方法，通过使用流体速度和电流密度（而不是速度和磁场）作为主要变量，避免了传统方法的一些固有困难。本文成功地应用速度-电流法证明了某些稳态MHD流动问题的适定性，并提出了一种求解数值逼近的有效有限元算法。提出的研究的一个目标是扩展速度-电流方法，分析和数值，以更复杂的固定问题，因为它们出现在液态金属技术和其他应用。主要的工具将是积分方程方法，混合变分公式，以及有限元和边界元方法的耦合。在另一个方向上，将适当推广速度-电流法来分析随时间变化的MHD流动问题。首先，重点将放在积分-微分方程后续系统的定性性质上，通过费奥多-伽辽金和半群方法；最终，我们的目标是设计一种有效的算法，用于在物理现实情况下的数值近似解。磁流体力学（或MHD）是研究导电流体与磁场宏观相互作用的理论。它对于控制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等许多工程问题都具有重要意义。它在地球物理学和天文学中也有应用，其中一个突出的例子是所谓的发电机问题，即地球磁场在液态金属地核中的起源问题。由于其实际意义，MHD流动问题长期以来一直是激烈的跨学科研究的主题，但除了相对简单的特殊情况外，对此类问题的严格数学和数值分析在很大程度上是未知领域。作者最近开发了一种新的方法来解决一类重要的MHD流动问题，这种方法绕过了传统分析的一些固有困难。本研究的目标是进一步研究这一方法，以设计定性和定量的方法，在物理现实情况下为MHD流动问题提供数学严谨和计算高效的解决方案。

9625096 Schmidt Magnetohydrodynamics (or MHD) is the theory of the macroscopic interaction of electrically conducting fluids with a magnetic field. In the viscous incompressible case, MHD flow is governed by the Navier-Stokes equations and the pre-Maxwell equations of the magnetic field. The latter will in general transcend the region of conducting fluid and, ideally, extend to all of space. It is mostly this feature, the electromagnetic interaction of the fluid with the outside world, which gives rise to challenging problems of mathematical analysis and numerical approximation. In earlier work, the authors have developed a novel approach to viscous incompressible MHD that avoids some intrinsic difficulties of traditional approaches by employing fluid velocity and current density (rather than velocity and magnetic field) as the primary variables. The velocity-current method has been successfully applied to prove the well-posedness of certain stationary MHD flow problems and to develop an efficient finite-element algorithm for the numerical approximation of solutions. One objective of the proposed research is to extend the velocity-current method, both analytically and numerically, to more complex stationary problems, as they arise in liquid-metal technology and other applications. Major tools will be integral-equation methods, mixed variational formulations, and the coupling of finite-element and boundary-element methods. In another direction, a suitable generalization of the velocity-current method will be employed to analyze time-dependent MHD flow problems. Initially, the focus will be on qualitative properties of the ensuing systems of integro-differential equations, via Faedo-Galerkin and semigroup methods; ultimately, the goal is again the design of efficient algorithms for the numerical approximation of solutions in physically realistic situations. Magnetohydrodynamics (or MHD) is the theory of the macroscopic interaction of el ectrically 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 one prominent example is the so-called dynamo problem, that is, the question of the origin of the Earth's magnetic field in its liquid metal core. Due to their practical relevance, MHD flow problems have long been the subject of intense cross- disciplinary research, but except for relatively simple special cases, the rigorous mathematical and numerical analysis of such problems is largely terra incognita. The authors have recently developed a novel approach to an important class of MHD flow problems that circumvents some intrinsic difficulties of traditional analyses. The objective of the proposed research is to further pursue this approach in order to design qualitative and quantitative methods for the mathematically rigorous and computationally efficient solution of MHD flow problems in physically realistic situations.