Mathematical Sciences: Non-linear Partial Differential Equations

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。在其物理解释中，该作品关注的是物质体的变形或演变。特别是，将努力了解压缩和旋转如何控制变形。例如，如果控制了无穷小的压缩，那么为了保证变形的平滑，还必须控制其他的旋转量吗？这个问题是速度场或瞬时变形场的拉格朗日版亥姆霍兹散度旋度分解，在几个必须计算大偏差的领域（计算塑性，气象科学中的锋面形成模型，不可压缩流的离散化）以及分配问题中自然出现，在这些问题中，人们试图将资源分配到一个新的分配中，从而使分配成本最小化。第二个主题是关于自由边界及其奇异扰动。在连续介质力学中，当一个快速转变发生时，模型假定存在一个尖锐的界面（火焰的燃烧，液体的凝固等）。将对陡变与连续现象的近似程度进行研究。例如，尖锐转变与窄频带连续转变的几何性质，或者当连续现象的极限实际上是预期的尖锐转变时，将进行分析。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

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. ***