Solution formula for generalized two-phase Stokes equations and its applications to maximal regularity; model problems

TL;DR

This paper derives a solution formula for the two-phase Stokes equations with surface tension and gravity, enabling resolvent and maximal regularity estimates without restrictive normal component assumptions.

Contribution

It reconstructs the solution formula to prove resolvent and maximal regularity estimates, relaxing previous normal component assumptions using an $H^$ calculus approach.

Findings

Established a solution formula for two-phase Stokes equations in the whole space.

Proved resolvent and maximal regularity estimates for the equations.

Weakened assumptions on normal components compared to previous work.

Abstract

In this paper we give a solution formula for the two-phase Stokes equations with and without surface tension and gravity in the whole space with flat interface. The solution formula has already considered by Shibata-Shimizu. However we reconstruct the formula so that we are able to prove resolvent estimate and maximal regularity estimate. In the previous work, they needed to assume additional conditions on normal components. We also take care of normal components, while the assumption becomes weaker than before. The method is based on an $H^\infty$ calculus which has already used for the Stokes problems with various boundary conditions in the half space.