Mathematical Sciences: Non-linear Partial Differential Equations

Caffarelli 9401168这个项目继续对涉及非线性偏微分方程的几个广泛领域进行数学研究。在两个中心主题中，一个涉及拉格朗日坐标和分配问题。在它的物理解释中，这项工作与物质物体的变形或演化有关。特别是，将努力了解压缩和旋转是如何控制变形的。例如，如果控制了无限小的压缩，还必须控制什么旋转量才能确保变形是平滑的？这个问题是速度或瞬时形变场的Helmholtz散度-旋度分解的拉格朗日版本，在必须计算大偏差(计算塑性、气象科学中的锋面形成模型、不可压缩流动的离散化)以及分配问题的几个领域中，自然地出现了这个问题，在这些领域，人们试图将资源分配到一个新的分配中，以使分配成本最小化。第二个主题与自由边界及其奇异扰动有关。在连续介质力学中，当发生快速转变时，模型假定存在尖锐的界面相(火焰燃烧、液体凝固等)，就会出现这些问题。将研究这种急剧转变与连续现象的接近程度。例如，尖锐跃迁相对于连续跃迁的窄带的几何性质，或者当连续现象的极限实际上是预期的急剧跃迁时，将被分析。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括对关于唯一性、光滑性和广度的问题的回答。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。***

Caffarelli 9401168 This project continues mathematical research on several broad areas involving nonlinear partial differential equations. Of the two central themes, one concerns Lagrangian coordinates and allocation problems. In its physical interpretation, the work is concerned with the deformation or evolution of a material body. In particular, efforts will be made to understand how compression and rotation control the deformation. For instance, if the infinitesimal compression is controlled, what other quantity of the rotation must be controlled to assure that the deformation be smooth? This question, a Lagrangian version of the Helmholtz divergence-curl decomposition for a velocity or instantaneous deformation field, has arisen naturally in several areas where one must compute large deviations (computational plasticity, models for front formation in meteorological sciences, discretization of incompressible flows) as well as problems of allocation, where one tries to allocate resources into a new distribution which minimizes the allocation costs. A second theme concerns free boundaries and their singular perturbations. These arise in continuum mechanics when a rapid transition takes place in which the model assumes the presence of a sharp interphase (the burning of a flame, the solidification of a liquid, etc.). Studies of how well the sharp transition approximates the continuous phenomena will be carried out. For instance, the geometric properties of the sharp transition versus a narrow band of continuous transition, or when the limit of the continuous phenomena is actually the expected sharp transition will be analyzed. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and gr owth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***