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

In this paper, we give a solution formula for the two-phase Stokes equations with and without surface tension and gravity over the whole space with a flat interface. The solution formula has already been considered by Shibata and Shimizu. However, we have reconstructed the formula so that we are able to easily prove resolvent and maximal regularity estimates. The previous work required the assumption of additional conditions on normal components. Here, although we consider normal components, the assumption is weaker than before. The method is based on an $H^{\mathrm{\infty }}$-calculus which has already been applied for the Stokes problems with various boundary conditions in the half-space.