AI 中文总结
本文研究带表面张力的两相Brinkman问题，证明其圆形稳态界面的渐近稳定性，通过带时间激活指数权重的Wiener型代数得到高度正则解的存在唯一性，初始扰动可指数衰减。
AI 中文摘要
我们研究如下系统：$\boldsymbol{R}^2$中有界单连通区域内的流体被另一具有锐利界面的流体包围，二者为充满恒定渗透率多孔介质的不可压缩Brinkman流，粘度相同，通过界面上的表面张力相互作用。我们假设速度在界面处无跳跃且在无穷远处衰减。我们建立了圆形界面（本系统的稳态解）的渐近稳定性，渐近稳定性所需初始扰动大小的技术阈值可明确计算。我们进一步证明初始扰动呈指数衰减。通过将扰动变量置于带时间激活指数权重的Wiener型代数中，我们证明了高度正则解的存在性、唯一性及对初始数据的连续依赖性，该代数允许具有更粗糙初始数据的解析解，且解包含在本问题临界标度指数的Wiener型代数中。
英文摘要
We study a system in which a fluid occupying a bounded simply connected region in $\mathbb{R}^{2}$ is surrounded by another fluid with sharp boundary. They are incompressible Brinkman flows of the same viscosity saturating a porous medium with constant permeability. They interact via surface tension on their interface. We assume that the velocity has no jump across the interface and decays at infinity. We establish the asymptotic stability of the circular interface, which is a steady-state solution to our system. The technical threshold for the size of the initial perturbation for asymptotic stability can be explicitly calculated. We further show that the initial perturbation decays exponentially. We prove the existence, uniqueness, and continuous dependence on initial data of highly regular solutions by containing the perturbation variable in a Wiener-type algebra with a time-activated exponential weight, which allows for analytic solutions with much coarser initial data. The solution is contained in the Wiener-type algebra with the critical scaling exponent for our problem.
Comments34 pages
