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

主要研究人员将研究几个相关的 非线性偏微分的渐近极限 流体力学中的方程。兴趣集中 在某些条件下， 方程变得越来越微不足道。 为 这些术语对应于 耗散和分散的物理效应。是 计划在两个复杂的层次上工作。第一， 更简单的层次是波的模型方程 非线性、色散和耗散的传播 所有的影响都存在。定性和定量 将寻求信息。获得的信息 将产生有助于模拟近岸的信息 区域进程。在更复杂的层面上， 来研究纳维尔-斯托克斯方程的各种极限 方程，包括无粘极限， 有界区域上的Navier-Stokes方程 边界，以及具有周期的统计解 边界条件（或所有空间）。也是 旨在研究纳维尔-斯托克斯方程 在一个通道中提出的方程是由耗散建模 非线性波动方程 目前的奖项将支持研究几个有趣的 和数学模型的重要渐近极限。的 这里考虑的各种限制出现在各种 物理学、机械学、海洋学、材料科学、 在生物学和其他地方，当偏微分方程 被用作模型。这里考虑的问题来自 主要是从流体力学，但就目前而言， 在我们的计划中取得成功，有一个更广泛的隐含范围。 研究中的零限值与以下因素有关： 流体运动中的耗散和分散。 以各种 在建模情况下，这些效应中的一个或另一个可以 忽视 那么问题就来了， 近似值是合理的，如果是，在什么流动条件下 以及时间尺度。 这些问题是 理论和实践的重要性，因为近似 等式通常更容易使用。该奖项将支持工作， 旨在解决的问题，包括基本点和方面 在使用这些方程来描述 真实的现象。

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.