Maximal Lp-Lq regularity for two-phase fluid motion in the linearized Oberbeck-Boussinesq approximation

Abstract

This paper is concerned with the generalized resolvent estimate and the maximal Lp-Lq regularity of the linearized Oberbeck-Boussinesq approximation for unsteady motion of a drop in another fluid without surface tension, which is indispensable for establishing the well-posedness of the Oberbeck-Boussinesq approximation for the two incompressible liquids separated by a closed interface. We prove the existence of R-bounded solution operators for the model problems and the maximal Lp-Lq regularity for the system. The key step is to prove the maximal Lp-Lq 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.