Mathematical Sciences: Nonlinear Evolutions and the Calculusof Variations

93222617斯滕贝格 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连续介质力学模型中产生的能量。 物理背景包括超导性和相变。 在所有情况下，能量极小化的结构是一个主要的兴趣问题。 除此之外，还将研究与能量相关的梯度流。 流动由抛物型偏微分方程或方程组模拟，其解倾向于降低能量。 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，某些几何结构往往表征稳态的解决方案，如最小的表面或漩涡，并试图了解如何在演化的后期阶段，这种结构propagatein时间。 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。 这可能包括关于独特性，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。 许多方程都是通过引用物理学原理而产生的，这些物理学原理都是以最小化某些量（如能量）为措辞的。 由此产生的数学表达式是一个变分问题，其解由偏微分方程表示。 ***

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. ***