Time-decay estimates for linearized two-phase Navier-Stokes equations with surface tension and gravity

TL;DR

This paper establishes time-decay estimates for solutions to linearized two-phase Navier-Stokes equations with surface tension and gravity, assuming the lower fluid is heavier, using solution formulas from prior resolvent problem analysis.

Contribution

It provides new $L_p-L_q$ decay estimates for the linearized equations under stable density conditions, extending understanding of two-phase flow dynamics.

Findings

Proves time-decay estimates for linearized two-phase Navier-Stokes equations.

Uses solution formulas from prior resolvent analysis.

Addresses stability when the lower fluid is heavier.

Abstract

The aim of this paper is to show time-decay estimates of solutions to linearized two-phase Navier-Stokes equations with surface tension and gravity. The original two-phase Navier-Stokes equations describe the two-phase incompressible viscous flow with a sharp interface that is close to the hyperplane $x_N=0$ in the $N$-dimensional Euclidean space, $N \geq 2$. It is well-known that the Rayleigh-Taylor instability occurs when the upper fluid is heavier than the lower one, while this paper assumes that the lower fluid is heavier than the upper one and proves time-decay estimates of $L_p-L_q$ type for the linearized equations. Our approach is based on solution formulas, given by Shibata and Shimizu (2011), for a resolvent problem associated with the linearized equations.