In this paper we consider a slab of viscous incompressible fluid bounded above by a free boundary, bounded below by a flat rigid interface, and acted on by gravity. The unique equilibrium is a flat slab of quiescent fluid. It is well-known that equilibria are asymptotically stable but that the rate of decay to equilibrium depends heavily on whether or not surface tension forces are accounted for at the free interface. The aim of the paper is to better understand the decay rate by studying a generalization of the linearized dynamics in which the surface tension operator is replaced by a more general fractional-order differential operator, which allows us to continuously interpolate between the case without surface tension and the case with surface tension. We study the decay of the linearized problem in terms of the choice of the generalized operator and in terms of the horizontal cross-section. In the case of a periodic cross-section we identify a critical order of the differential operator at which the decay rate transitions from almost exponential to exponential.

中文翻译：

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具有分数边界算子的线性自由表面Navier-Stokes方程解的衰减

在本文中，我们考虑了一块粘性的不可压缩流体，其上自由边界为边界，下边界为平坦的刚性界面，并受重力作用。独特的平衡是静态流体的平坦平板。众所周知，平衡是渐近稳定的，但是衰减到平衡的速率在很大程度上取决于是否在自由界面处考虑了表面张力。本文的目的是通过研究线性化动力学的一般性来更好地理解衰减率，其中将表面张力算子替换为更通用的分数阶微分算子，这使我们能够在没有表面张力的情况下连续地在壳体之间进行插值和表面张力的情况。我们根据广义算子的选择和水平横截面研究线性化问题的衰减。在周期性横截面的情况下，我们确定了微分算子的临界阶数，在该阶数处，衰减率从几乎指数级过渡到指数级。