Decay of viscous surface waves without surface tension
Decay of viscous surface waves without surface tension
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DOI：
10.2140/apde.2013.6.1429
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发表时间：
2010-11
期刊：
影响因子：
--
通讯作者：
Yan Guo;Ian Tice
中科院分区：
文献类型：
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作者：
Yan Guo;Ian Tice
考虑空气下面有限深度的粘性流体。自Beale_1工作以来，在没有界面张力效应的情况下，小振幅自由表面的长时间行为一直是一个有趣的问题。在这本专著中，我们开发了一个新的数学框架来解决这个问题。如果自由界面是水平无限大的，我们建立它衰减到一个平面在一个代数率。另一方面，如果自由界面是周期性的，我们确定它几乎以指数速率衰减，即以任意快的代数速率衰减，该速率由数据的微小性决定。我们的框架包含了一些新的技术，包括：（1）移动边界下Navier-Stokes方程的局部适定性理论;（2）将高阶能量的有界性与低阶能量的衰减耦合起来的双层能量方法，后者是平衡自由界面最高导数增长所必需的;（3）控制负和正Sobolev范数，这增强了插值估计，并允许无限表面波的衰减;（4）与能量方法兼容的局部化过程，并允许在周期性情况下弯曲的下表面几何形状。我们的衰减结果导致建设的全球时间解决方案的表面波问题。
Consider a viscous fluid of finite depth below the air. In the absence of the surface tension effect at the air-fluid interface, the long time behavior of a free surface with small amplitude has been an intriguing question since the work of Beale \cite{beale_1}. In this monograph, we develop a new mathematical framework to resolve this question. If the free interface is horizontally infinite, we establish that it decays to a flat surface at an algebraic rate. On the other hand, if the free interface is periodic, we establish that it decays at an almost exponential rate, i.e. at an arbitrarily fast algebraic rate determined by the smallness of the data. Our framework contains several novel techniques, which include: (1) a local well-posedness theory of the Navier-Stokes equations in the presence of a moving boundary; (2) a two-tier energy method that couples the boundedness of high-order energy to the decay of low-order energy, the latter of which is necessary to balance out the growth of the highest derivatives of the free interface; (3) control of both negative and positive Sobolev norms, which enhances interpolation estimates and allows for the decay of infinite surface waves; (4) a localization procedure that is compatible with the energy method and allows for curved lower surface geometry in the periodic case. Our decay results lead to the construction of global-in-time solutions to the surface wave problem.