Local well-posedness for two-phase fluid motion in Oberbeck-Boussinesq approximation

In this talk, we survey some recent progress of the local well-posedness of solution of the free interface problem to the Oberbeck-Boussinesq approximation for the unsteady motion of a drop in another fluid separated by an unknown closed interface with surface tension. We shall establish the existence and uniqueness of solution of this nonlinear problem by following steps. Firstly, we transform the Oberbeck-Boussinesq approximation by using the Hanzawa transformation from the free interface problem to a fixed interface problem. Secondly, we prove the maximal L^p-L^q regularity for the system and the existence of ℛ-bounded solution operators for the model problems to be used. It is essential to reach the well-posedness of the solution to the Oberbeck-Boussinesq approximation. Thirdly, it is one of the key steps to prove the maximal L^p-L^q  regularity theorem for the linearized heat equation with the help of the ℛ-bounded solution operators for the corresponding resolvent problem and the Weis operator-valued Fourier multiplier theorem. Finally, using maximal L^p-L^q regularities for the two-phase fluid motion of the linearized system to establish the existence and uniqueness of the solutions of nonlinear problem with the help of the contraction mapping principle, in which the differences of nonlinear terms are estimated. It is based on the joint works with Wei Zhang.