Selected problems in applied mathematics

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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.