Mathematical Sciences: Non-linear Partial Differential Equations

Caffarelli 9401168 该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。 从物理学的角度来看，这部作品关注的是物质体的变形或演化。 特别地，将努力理解压缩和旋转如何控制变形。 例如，如果控制了无限小的压缩，那么还必须控制什么样的旋转量才能保证变形是平滑的？ 这个问题是速度场或瞬时变形场的亥姆霍兹发散-旋度分解的拉格朗日形式，在必须计算大偏差的几个领域中自然出现（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，其中试图将资源分配到最小化分配成本的新分布中。 第二个主题是关于自由边界及其奇异摄动。 这些在连续介质力学中出现，当发生快速转变时，模型假设存在尖锐的界面（火焰燃烧，液体凝固等）。 将进行关于急剧转变与连续现象的近似程度的研究。 例如，将分析急剧转变与连续转变的窄带的几何特性，或者连续现象的极限实际上是预期的急剧转变。 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建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于唯一性、平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 ***

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. ***