Div-curl systems and Variational principles

University Of Houston, Houston TX

Investigators

Abstract

This project will support the investigator's analysis of certain basic systems of equations of interest in science and engineering. Our primary topic will be boundary value problems for div-curl system on bounded regions in space. Both Maxwell's equations for an electromagnetic field and some problems in fluid mechanics are governed by this system with certain associated boundary conditions. There still are many open questions about these systems and their solutions. The proposed research centers on how to describe the mathematical properties, and approximation, of solutions of these boundary value problems. This information is needed for the development of good algorithms for the computation of these fields or flows -- and should have wide applicability for the numerical modeling of devices and equipment. Mathematically, these div-curl systems are over-determined systems of equations that only have solutions when certain compatibility conditions hold -- and often require extra data in addition to standard boundary conditions. These extra conditions include both natural analytical conditions and some subtle conditions that arise from geometrical considerations. They sometimes are known to experts on these subjects, but their statements have previously only been vaguely stated, since the precise versions require considerable geometrical and analytical detail. They have been carefully described in recent papers of the PI and his collaborators. This proposal is to support the implementation, and further development, of these results in some models that arise in applications. The proposed research will use the variational characterization of the solutions to obtain sharp results about the finite energy solutions of these systems in a number of important geometrical situations. We also intend to investigate the development of good methods for the numerical approximation of the solutions in various different geometrical configurations. A particular interest will be the analysis of the solution of certain models of ferromagnetic bodies and electrostatic discharges.