On the solution operators arising from the gas-liquid two-phase problem in unbounded domains with finite depth

TL;DR

This paper develops maximal regularity estimates for evolution equations from the gas-liquid two-phase Navier-Stokes problem in unbounded domains, using explicit resolvent operators and R-solvers.

Contribution

It constructs the R-solver for the resolvent problem, providing explicit analysis of the gas-liquid operator in unbounded domains with flat boundaries.

Findings

Established maximal $L_p$-$L_q$ regularity estimates.

Constructed explicit resolvent operators for the two-phase problem.

Analyzed the resolvent operators in unbounded domains with flat boundaries.

Abstract

This paper studies some evolution equations arising from the sharp interface problem of the compressible-incompressible Navier-Stokes equations in unbounded domains in $\mathbb{R}^N (N\geq2)$, where the viscous gases initially occupy the upper half space and the viscous liquids below initially lie in the strip-like domain. In order to establish the maximal $L_p$-$L_q$ regularity estimates of the evolution problem, we construct the R-solver of the resolvent problem associated to the gas-liquid two-phase operator. The crucial part of our proof lies in the analysis of the explicit resolvent operators defined in the unbounded domains with flat boundaries.