Mathematical Sciences: Nonlinear Evolutions and the Calculusof Variations

93222617斯特恩伯格该奖项支持对涉及非线性偏微分方程和变分法的一系列问题的数学研究。大部分工作的基础通常是从连续介质力学的模型中产生的能量。物理背景包括超导和相变。在所有情况下，能量极小器的结构都是一个主要感兴趣的问题。除此之外，还将对与能量相关的梯度流进行研究。流动由抛物型偏微分方程组或其解倾向于降低能量的系统来建模。一个主要目标是描述这些系统在努力达到平衡状态时的大时间行为。在所有情况下，某些几何结构倾向于描述稳态解，如极小表面或旋涡，人们试图了解这些结构如何在演化的后期阶段传播。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。这可能包括回答有关唯一性、平稳性和成长性的问题。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。许多方程式是通过引用物理原理而产生的，这些原理的表述是最小化某些量，如能量。由此得到的数学表达式是一个变分问题，其解由偏微分方程式表示。***

93222617 Sternberg This award supports mathematical research on a range of problems involving nonlinear partial differential equations and the calculus of variations. The basis for much of the work is typically an energy arising from a model in continuum mechanics. Physical contexts include superconductivity and phase transitions. In all cases, the structure of minimizers of the energy is an issue of primary interest. Beyond this, work will be done on gradient flows associated with the energy. The flows are modeled by parabolic partial differential equations or systems whose solution tend to decrease the energy. A primary goal is to characterize the large-time behavior of these systems as they strive to achieve an equilibrium state. In all cases, certain geometric structures tend to characterize steady-state solutions such as minimal surfaces or vortices and one seeks to understand how such structures propagatein time during the late stages of the evolution. 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 growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. Many equations arise by invoking physical principles, phrased in terms of minimizing certain quantities, such as energy. The resulting mathematical expression is a variational problems whose solution is expressed by a partial differential equation. ***