论文标题
多层无压制的Navier-Stokes方程的行驶波解决方案
Traveling wave solutions to the multilayer free boundary incompressible Navier-Stokes equations
论文作者
论文摘要
对于自然数量$ m \ ge 2 $，我们研究了$ m $层的有限深度，水平无限，粘性和不可压缩的液体，下面是平坦的刚性底部。相邻的层在自由接口区域相交，顶层也由自由边界界定。一个均匀的重力场（正常与刚性底部）作用在流体上。我们假设流体质量密度严格从底部到顶部降低，并考虑在自由表面上作用的情况下有和没有表面张力。除了这些重力毛细血管效应外，我们还允许力对散装和外部应力张量作用，以对自由界面区域作用。这两个额外的力都处于行动波形形式：与时间无关的坐标系统以恒定的非平凡速度移动时，平行于下刚性边界。在二维流体的情况下，没有表面张力，并且在较高尺寸的情况下所有正面张力都没有表面张力，我们证明，对于每种足够小的力和应力元组，都有一个波动波溶液。最近建立了一层配置（$ M = 1 $）的行驶波解决方案的存在，据我们所知，本文是第一次在$ m $ aylayer布置中对不可压缩的Navier-Stokes方程进行的第一次构造。
For a natural number $m \ge 2$, we study $m$ layers of finite depth, horizontally infinite, viscous, and incompressible fluid bounded below by a flat rigid bottom. Adjacent layers meet at free interface regions, and the top layer is bounded above by a free boundary as well. A uniform gravitational field, normal to the rigid bottom, acts on the fluid. We assume that the fluid mass densities are strictly decreasing from bottom to top and consider the cases with and without surface tension acting on the free surfaces. In addition to these gravity-capillary effects, we allow a force to act on the bulk and external stress tensors to act on the free interface regions. Both of these additional forces are posited to be in traveling wave form: time-independent when viewed in a coordinate system moving at a constant, nontrivial velocity parallel to the lower rigid boundary. Without surface tension in the case of two dimensional fluids and with all positive surface tensions in the higher dimensional cases, we prove that for each sufficiently small force and stress tuple there exists a traveling wave solution. The existence of traveling wave solutions to the one layer configuration ($m=1$) was recently established and, to the best of our knowledge, this paper is the first construction of traveling wave solutions to the incompressible Navier-Stokes equations in the $m$-layer arrangement.
