Zero-Dissipation and Zero-Dispersion Limits Arising in Fluid Mechanics

首席研究员将研究流体力学中出现的非线性偏微分方程解的几个相关的渐近极限。人们的兴趣集中在解在方程中的某些项变得越来越微不足道的极限情况下如何表现。对于这里所看到的情况，这些术语对应于耗散和弥散的物理效应。它计划在两个复杂程度上工作。第一个层次是波传播的模型方程，它考虑了非线性、色散和耗散效应。将寻求定性和定量两方面的信息。所获得的信息将产生有助于模拟近岸带过程的信息。在更复杂的水平上，计划研究Navier-Stokes方程的各种极限，包括具有固定边界的有界域中的Navier-Stokes方程的无粘性极限，以及具有周期边界条件的统计解(或整个空间)的无粘性极限。本文还将研究耗散非线性波动方程对航道中的Navier-Stokes方程的模拟效果。目前的奖项将支持对数学模型的几个有趣和重要的渐近极限的研究。这里考虑的那种限制出现在物理、力学、海洋学、材料科学、生物学和其他使用偏微分方程作模型的各个领域。这里考虑的问题主要来自流体力学，但就我们在程序中的成功而言，还有更广泛的隐含范围。研究中的零极限值与流体运动中的耗散和弥散有关。在各种建模情况下，可以忽略这些影响中的一种或另一种。于是问题出现了，这样的近似是否合理，如果是，在什么流动条件下，在什么时间尺度上。这些问题具有重要的理论和实践意义，因为近似方程往往更易于使用。该奖项将支持旨在解决问题的工作，这些问题包括在使用这些方程描述真实现象时出现的基本点和方面。

The Principal Investigators will study several related asymptotic limits of nonlinear partial differential equations arising in fluid mechanics. Interest is focused on how solutions behave in limits where certain terms in the equations become increasingly negligible. For the situations in view here, these terms correspond to the physical effects of dissipation and dispersion. It is planned to work at two levels of complexity. The first, and easier level is that of model equations for wave propagation where nonlinear, dispersive and dissipative effects are all present. Both qualitative and quantitative information will be sought. The information obtained will yield information helpful to modelling near-shore zone processes. At a more complex level, it is planned to investigate various limits of the Navier-Stokes equations including the inviscid limit for the Navier-Stokes equations in bounded domains with fixed boundaries, and for statistical solutions with periodic boundary conditions (or in all of space). It is also intended to investigate how well the Navier-Stokes equations posed in a channel are modelled by dissipative nonlinear wave equations. The present award will support research on several interesting and important asymptotic limits for mathematical models. The kind of limits under consideration here arise in various areas of physics, mechanics, oceanography, materials science, biology, and elsewhere when partial differential equations are used as models. The problems considered here derive principally from fluid mechanics, but in so far as we are successful in our program, there is a broader implied scope. The zero-limits under study are those associated with dissipation and dispersion in fluid motion. In various modeling situations, one or the other of these effects may be ignored. The question then arises whether or not such approximations are justified and, if so, under what flow conditions and over what time scales. These questions are of theoretical and practical importance since the approximating equations are often easier to use. This award will support work that aims to address issues including fundamental points and aspects that arise in the use of these equations as descriptions of real phenomena.