Asymptotic behavior of solutions for the thermoviscous acoustic systems

TL;DR

This paper analyzes the long-term behavior of solutions to thermoviscous acoustic systems, deriving optimal estimates and asymptotic profiles, and demonstrating the inviscid limit as momentum diffusion vanishes.

Contribution

It provides a rigorous analysis of the asymptotic properties and inviscid limits of thermoviscous acoustic systems using WKB, Fourier, and energy methods.

Findings

Wave effects dominate in lower dimensions.

Optimal estimates and asymptotic profiles are derived.

Global inviscid limits are rigorously established.

Abstract

We study some asymptotic properties of solutions for the acoustic coupled systems in thermoviscous fluids which was proposed by [Karlsen-Bruus, \emph{Phys. Rev. E} (2015)]. Basing on the WKB analysis and the Fourier analysis, we derive optimal estimates and large time asymptotic profiles of the energy term via diagonalization procedure, and of the velocity potential via reduction methodology. We found that the wave effect has a dominant influence for lower dimensions comparing with thermal-viscous effects. Moreover, by employing suitable energy methods, we rigorously demonstrate global (in time) inviscid limits as the momentum diffusion coefficient vanishes, whose limit model can be regarded as the thermoelastic acoustic systems in isotropic solids. These results explain some influence of the momentum diffusion on asymptotic behavior of solutions.