9424448 Xu Many industrial processes involve the electrical heating of a conductor. A typical example is the electrical spot welding of metal sheets in the manufacturing of such products as automobile bodies, tanks, and armored vehicles. The mathematical modeling of such processes often leads to systems of degenerate partial differential equations with diffusive and convective couplings. The study of these systems leads to deep and intriguing questions in nonlinear analysis. In these considerations, the presence of strong nonlinearities and degeneracy gives rise to very rich structures of solutions. Our recent research findings indicate that solutions of these systems display new phenomena that cannot be incorporated into the classical variational formulation, and a new notion of a capacity solution is, therefore, introduced to study these systems. In this proposal, we propose to study the finer properties of capacity solutions, such as regularity, partial regularity, large time behavior, and other related questions. The proposed research is important from the point of view of industrial applications. The mathematical results we wish to obtain can lead to a more efficient production line, a new or better product. For example, in the welding of metal sheets, one of our objectives is to find out how the welding time is related to the material properties and the current applied. This information is critical to the automation of the welding procedure.
