Mathematical Sciences: On a Class of Degenerate Systems Arising from the Electrical Heating of Conductors

许多工业过程都涉及到导体的电加热。一个典型的例子是制造汽车车身、坦克和装甲车等产品的金属板的电点焊。这类过程的数学建模常常导致具有扩散和对流耦合的退化偏微分方程系统。对这些系统的研究导致了非线性分析中深刻而有趣的问题。在这些考虑中，强非线性和简并性的存在产生了非常丰富的解结构。我们最近的研究结果表明，这些系统的解显示出经典变分公式无法包含的新现象，因此引入了容量解的新概念来研究这些系统。在本提案中，我们建议研究容量解的更精细性质，如正则性，部分正则性，大时间行为以及其他相关问题。从工业应用的角度来看，本研究具有重要意义。我们希望得到的数学结果可以使我们设计出更高效的生产线，生产出新的或更好的产品。例如，在金属板的焊接中，我们的目标之一是找出焊接时间与材料性能和施加的电流之间的关系。这些信息对焊接过程的自动化至关重要。

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.