论文标题
向倾斜或周期性的无压边界纳维尔 - 长方体方程的行驶波解决方案
Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations
论文作者
论文摘要
本文涉及为自由边界不可压缩的Navier-Stokes系统的构建。我们研究了一个粘性域中的单层粘性液体，该结构域由平坦的刚性表面及以上由移动的表面界定。流体由散装力和表面应力作用于平行于流体底部移动的坐标系中。我们还假设流体受到均匀的重力，可以将其分解为垂直分量的总和，而分量则位于行动波速度的方向上。例如，在倾斜平面的流体流量的建模中，出现了这种配置。我们还通过允许流体横截面在各个方向上进行周期性来研究周期性的效果。重力场的水平成分产生了纯粹的剪切流的固定溶液，我们通过隐式函数参数构建解决方案作为扰动。我们分析的一个重要组成部分是开发各向异性Sobolev空间的一些新功能分析性能，包括这些空间是超临界状态中的代数，这可能具有独立的利益。
This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. The fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom. We also assume that the fluid is subject to a uniform gravitational force that can be resolved into a sum of a vertical component and a component lying in the direction of the traveling wave velocity. This configuration arises, for instance, in the modeling of fluid flow down an inclined plane. We also study the effect of periodicity by allowing the fluid cross section to be periodic in various directions. The horizontal component of the gravitational field gives rise to stationary solutions that are pure shear flows, and we construct our solutions as perturbations of these by means of an implicit function argument. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces, including that these spaces are an algebra in the supercritical regime, which may be of independent interest.
