Mathematical Sciences: Viscous Incompressible Magnetohydrodynamics: Analysis and Numerical Approximation

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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.