Global Solvability of the Problem on Two-Phase Capillary Fluid Motion in the Oberbeck–Boussinesq Approximation

Abstract

Unsteady motion of a drop in another incompressible fluid bounded by a rigid surface is considered in the Oberbeck–Boussinesq approximation. The liquids are separated by a closed unknown interface \(\Gamma _t\) where surface tension is taken into account. Global existence theorem for the problem is stated in Holder classes of functions provided that the data have small norms and the initial configuration of the drop is close to a ball with the center in drop’s barycenter. It is shown that velocity vector field and temperature deviation decay exponentially as \(t\rightarrow \infty \), the interface between the liquids tending to a sphere \(\{|x-h_\infty |= R_0\}\) with a center \(h_\infty \), the limiting position of drop’s barycenter. It is established that if the initial data are small enough, the inner liquid will remain strictly inside the other one during all the time. The proof is based on the exponential estimate of a generalized energy and on a local existence theorem of the problem in anisotropic Holder spaces.