Regularity of Degenerate Partial Differential Equations and Applications

简并偏微分方程的正则性及应用

• 批准号: 341250-2012
• 负责人: Rios, Cristian
• 金额: $ 1.09万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572210
• 关键词: Regularity; Degenerate; Partial; Differential; Equations

Partial Differential Equations (PDEs) is the mathematical language science uses to model a wide variety of processes like heat flow, wave propagation, biological functions in humans and animals, financial markets, celestial dynamics, and the engineering of structures, among many other processes. Most scientific principles and theories can be elegantly stated using differential equations. For this reason PDEs have been extensively studied ever since Newton and Leibniz set the foundations for differential Calculus, and today there is a good understanding of what is now called classical theory of PDEs. However, many applications do not conform to the restrictions of classical models, as the equations governing their nature are ''degenerate'' in some aspects. Consequently, the objective of this proposal is to investigate key properties of solutions to PDEs with different types of degeneracies. The problems considered lie on the fringe of known theory, but escape classical techniques for their treatment. The importance of exploring these frontiers resides on the deeper insights gained for the general theory of PDEs and on the newly developed techniques that will offer potential for further applications.

偏微分方程（PDE）是一种数学语言，科学用来模拟各种各样的过程，如热流，波传播，人类和动物的生物功能，金融市场，天体动力学和结构工程，以及许多其他过程。大多数科学原理和理论都可以用微分方程优雅地表述。由于这个原因，自从牛顿和莱布尼茨为微分学奠定基础以来，偏微分方程就被广泛研究，今天人们对偏微分方程的经典理论有了很好的理解。然而，许多应用不符合经典模型的限制，因为控制其性质的方程在某些方面是“退化”的。 因此，这个建议的目的是调查的关键属性的解决方案，以不同类型的退化偏微分方程。所考虑的问题处于已知理论的边缘，但却逃脱了经典技术的处理。探索这些前沿领域的重要性在于对偏微分方程一般理论的更深层次的认识，以及新开发的技术，这些技术将为进一步的应用提供潜力。