Nonlinear Wave Equation Asymptotics and Functions of Bounded Higher Variation

本建议集中在两个不同但相关的研究方向：非线性波动方程的奇异极限，以及函数空间BnV的理论和应用的发展。前一个问题需要描述包含一个小参数的非线性波动方程的某些族的渐近行为。这些大多是数学物理中著名的方程。具体问题包括与Gross-Pitaevsky方程解中的涡动力学有关的一些问题，例如三维或多维涡旋细丝的动力学。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由精确意义上具有有界维变化的函数组成。首席研究员将研究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。我们提出了包括涡丝在某些超流体中的运动在内的一些问题，以及纯数学分析中的相关问题。漩涡灯丝最常见的例子是烟圈。这样的环开始时很小，随着它们的移动，它们的直径往往会扩大。同时，随着涡旋度的降低，它们的旋转速度也会变慢。如果在（玻色子）液氦中吹一个烟圈，它会表现出完全不同的行为。在液氦中，涡度只能取一定的值。这实际上可以防止沿烟圈的涡度减小，从而防止烟圈膨胀。因此，预计液氦中的涡旋丝在不增加长度和不改变旋转速率的情况下传播。一些被提议的工作包括从描述玻色子超流体的基本方程开始，发展对这种细丝运动的理解。这很困难，部分原因是涡旋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.