Nonlinear Wave Equation Asymptotics and Functions of Bounded Higher Variation

这项建议集中在两个不同但相关的研究领域：非线性波动方程的奇异极限和函数空间BnV的理论和应用的发展。前一个问题需要描述某些包含小参数的非线性波动方程的渐近行为。这些方程大多是来自数学物理的众所周知的方程。具体问题包括在Gross-Pitaevsky方程的解中与涡旋动力学有关的一些问题，例如三维或更多维的涡丝动力学。功能空间的BNV道岔为这一问题和相关问题提供了一个自然的环境。BnV由精确意义上有界的维变差函数组成。主要研究人员将研究BNV函数的性质，着眼于在偏微分方程组和变分问题中的应用。我们建议研究一些问题，包括某些类型的超流体中涡丝的运动，以及纯数学分析中的相关问题。最常见的涡丝的例子是烟环。这种环一开始很小，随着它们的移动，它们的直径有扩大的趋势。同时，随着涡量的减少，它们的自转速度也会变得更慢。如果一个人能在(玻色子)液氦中吹出一个烟圈，它将表现出完全不同的行为。在液氦中，涡度只能假定一定的值。这实际上可以防止沿烟环的涡量减小，进而防止烟环膨胀。因此，液氦中的涡旋细丝有望在不增加其长度和不改变旋转速度的情况下传播。一些拟议的工作涉及从描述玻色子超流的基本方程开始，发展对这种细丝运动的理解。这是困难的，部分原因是涡流细丝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用于描述这些涡旋细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.