Nonlinear Wave Equation Asymptotics and Functions of Bounded Higher Variation

该建议侧重于两个不同的，但相关的研究线：奇异极限的非线性波动方程，和发展oftheory和应用的函数空间，BnV。前一个问题需要描述的渐近行为的某些家庭的非线性波动方程包含一个小参数。这些大多是著名的方程从数学物理。具体问题包括一些问题有关的涡dynamicsin的解决方案的Gross-Pitaevsky方程，例如动态ofvortex filaments在三维或更多维。函数空间BnV为这个问题和相关问题提供了一个自然的环境。BnV由函数组成，在精确的意义上，具有有界的n维变化。主要研究者将研究BnV函数的性质，着眼于在偏微分方程和变分法问题中的应用。我们建议研究一些问题，包括某些超流体中涡丝的运动，以及纯数学分析中的相关问题。涡丝最常见的例子是烟圈。这种环开始很小，当它们运动时，它们的直径趋向于扩张。同时，随着它们的涡度减小，它们旋转得更慢。 如果人们能在（玻色子）液氦中吹出烟圈，它会表现出完全不同的行为。 在液氦中，涡量只能取一定的值。这实际上可以防止沿着烟圈的涡度减小，从而防止烟圈膨胀。因此，液氦中的涡丝预计在其长度不增加和旋转速率不改变的情况下传播。一些研究工作涉及到从描述玻色子超流体的基本方程出发来理解这种涡丝运动。这是困难的，部分原因是涡丝并没有以任何明确的方式出现在方程中。该项目的其他方面包括研究一类可以用来描述这些涡丝和其他相关现象的函数。

This proposal focuses on two distinct but related lines of research:singular limits of nonlinear wave equations, and the development oftheory and applications of a function space, BnV.The former problems entail describing the asymptotic behavior of certainfamilies of nonlinear wave equations containing a small parameter.These are mostly well-known equations from mathematical physics.Specific problems include a number of issues related to vortex dynamicsin solutions of the Gross-Pitaevsky equation, for example dynamics ofvortex filaments in three or more dimensions. The function space BnV turnsout to provide a natural setting for this and related problems. BnVconsists of functions that, in a precise sense, have boundedn-dimensional variation. The principal investigator will investigateproperties of BnV functions with an eye towards applications to problemsin partial differential equations and the calculus of variations.We propose to study some questions that include the motion of vortexfilaments in certain kinds of superfluids, and relatedquestions in pure mathematical analysis.The most familiar example of a vortex filament is a smoke ring.Such rings start out small, and as they move, their diameters tendto expand. Simultaneously, they rotate more slowly as their vorticitydecreases. If one could blow a smoke ring in (bosonic) liquid helium,it would exhibit quite different behavior. In liquid helium,the vorticity can only assume certain values. This in effect shouldprevent the vorticity along a smoke ring from decreasing, which inturn should prevent the ring from exanding. Thus a vortex filament inliquid helium is expected to propagate without its length increasing andwithout the rate of rotation changing. Some of the proposed workinvolves developing an understanding of this filament motion,starting from basic equations that describe bosonic superfluids.This is difficult, in part because the vortex filaments do notappear in the equations in any explicit way. Other aspectsof the project involve studying a class of functions that canbe used to describe these vortex filaments and other relatedphenomena.