Critical global well-posedness for the two-phase Brinkman problem with surface tension

Abstract

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.