Global well-posedness for two-phase fluid motion in the Oberbeck–Boussinesq approximation

This paper focuses on the global well-posedness of the Oberbeck–Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial state of the drop is close to a ball B_R with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of B_R. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck–Boussinesq approximation in a fixed domain. From there, we establish time-weighted estimates of solutions for the shifted equation using maximal L^p–L^q regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang [J. Differ. Equations 322, 101–134 (2022)]. Using time decay estimates of the semigroup, we then obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval (0, T) was proven by Hao and Zhang [Commun. Pure Appl. Anal. 22(7), 2099–2131 (2023)]. After that, we demonstrate that the solutions can be extended beyond T by analyzing the properties of the roots of algebraic equations.