Optimal decay rates of a non-conservative compressible two-phase fluid model

TL;DR

This paper establishes the optimal decay rates for solutions to a non-conservative compressible two-phase fluid model in three-dimensional space, matching heat equation decay rates and providing a comprehensive analytical framework.

Contribution

It introduces a general framework to derive optimal decay rates for solutions and their derivatives, extending previous work with new spectral and energy analysis techniques.

Findings

Derived optimal decay rates matching heat equation behavior.

Established lower bounds on decay rates for specific initial data.

Developed a unified analytical approach using spectral and energy methods.

Abstract

We are concerned with the time decay rates of strong solutions to a non-conservative compressible viscous two-phase fluid model in the whole space R3. Compared to the previous related works, the main novelty of this paper lies in the fact that it provides a general framework that can be used to extract the optimal decay rates of the solution as well as its all-order spatial derivatives from one-order to the highest-order, which are the same as those of the heat equation. Furthermore, for well-chosen initial data, we also show the lower bounds on the decay rates. Our methods mainly consist of Hodge decomposition, low-frequency and high-frequency decomposition, delicate spectral analysis and energy method based on finite induction.