论文标题
两流体表面张力系统的反馈稳定，该系统在低雷诺数下建模肥皂泡的运动：二维情况
Feedback stabilization of a two-fluid surface tension system modeling the motion of a soap bubble at low Reynolds number: The two-dimensional case
论文作者
论文摘要
本文的目的是设计一个反馈操作员，以在无限时间范围内稳定一个系统，该系统模拟了粘性不可压缩的流体与肥皂泡的变形之间的相互作用。后者由一个接口表示，将$ \ mathbb {r}^2 $的有界域分隔为两个连接的零件，这些零件充满了粘性不可压缩的流体。该界面是1个球体的平滑扰动，周围的流体满足了时间相关域中不可压缩的Stokes方程。表面的平均曲率定义了表面张力力，该表面张力力诱导cauchy应力张量的正常痕迹的跳跃。流体的响应是界面上的速度轨迹，通过速度的相等性来管理后者的时间演变。假定数据足够小，特别是初始扰动，即肥皂泡的初始形状与圆足够近。控制函数是界面上的表面张力类型力。我们将其设计为两个反馈操作员的总和：一个是显式的，第二个是有限维度的。它们使我们能够定义一个控制操作员，该控制器将肥皂气泡稳定到具有任意指数衰减速率的圆圈，直至翻译，并与外边界无接触。
The aim of this paper is to design a feedback operator for stabilizing in infinite time horizon a system modeling the interactions between a viscous incompressible fluid and the deformation of a soap bubble. The latter is represented by an interface separating a bounded domain of $\mathbb{R}^2$ into two connected parts filled with viscous incompressible fluids. The interface is a smooth perturbation of the 1-sphere, and the surrounding fluids satisfy the incompressible Stokes equations in time-dependent domains. The mean curvature of the surface defines a surface tension force which induces a jump of the normal trace of the Cauchy stress tensor. The response of the fluids is a velocity trace on the interface, governing the time evolution of the latter, via the equality of velocities. The data are assumed to be sufficiently small, in particular the initial perturbation, that is the initial shape of the soap bubble is close enough to a circle. The control function is a surface tension type force on the interface. We design it as the sum of two feedback operators: one is explicit, the second one is finite-dimensional. They enable us to define a control operator that stabilizes locally the soap bubble to a circle with an arbitrary exponential decay rate, up to translations, and up to non-contact with the outer boundary.
