Selected problems in applied mathematics

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该研究项目是对光学，流体动力学，人口动力学，分子动力学和热力学中出现的偏微分方程类的解的行为进行分析和数值研究。 正在研究的问题领域包括分析目前分子动力学算法的有效性和局限性;质量临界和能量临界非线性薛定谔方程的爆破;三维不可压缩纳维尔-斯托克斯方程解的有限时间爆破研究，并扩展到其他流体力学方程，包括准地转方程;以及种群动力学和群集过程中聚集模型的连续极限和适定性，这些研究在物理学、生物学和工程学中有重要的应用。 例如，立方非线性薛定谔方程在低温物理中被用来描述玻色-爱因斯坦凝聚。 三维Navier-Stokes方程和地面准地转方程在天气预报和地球大气环流模拟中有着重要的应用。 分子动力学算法的数学分析将有助于解释在真实的材料的计算机模拟中观察到的许多现象，对于这些现象，实验方法往往是无法接近的。 对聚集方程的分析可以对种群动力学和生物学中用于研究群集行为的模型进行公正的评价。 本研究的主要目的是为这些问题开发一个合适的数学框架。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This research project is an analytical and numerical study of the behavior of solutions to classes of partial differential equations that arise in optics, fluid dynamics, population dynamics, molecular dynamics, and thermodynamics. The problem areas under study include analysis of the validity and limitations of current molecular dynamics algorithms; blow up for mass-critical and energy-critical nonlinear Schrodinger equations; investigation of finite-time blow-up in solutions to the three-dimensional incompressible Navier-Stokes equations, with extension to other hydrodynamics equations, including the quasi-geostrophic equations; and the continuum limit and well-posedness of aggregation models arising in population dynamics and swarming.This research has important applications in physics, biology, and engineering. For example, the cubic nonlinear Schrodinger equation is used to describe Bose-Einstein condensates in low temperature physics. The three-dimensional Navier-Stokes equations and the surface quasi-geostrophic equations have applications in weather prediction and modeling the earth's atmospheric circulation. The mathematical analysis of molecular dynamics algorithms will help to explain many observed phenomena in computer simulations of real materials, for which experimental methods are often inaccessible. The analysis of the aggregation equations can give a candid assessment of models that are used in studying the swarming behavior in population dynamics and biology. The main objective of this research is to develop a suitable mathematical framework for these problems.