The inviscid limit of third-order linear and nonlinear acoustic equations

TL;DR

This paper investigates the inviscid limit of third-order linear and nonlinear acoustic equations in thermally relaxing fluids, demonstrating convergence of solutions to inviscid models and supporting findings with numerical experiments.

Contribution

It provides a rigorous analysis of the inviscid limit for third-order acoustic equations, including nonlinear models, with convergence rates and numerical validation.

Findings

Solutions converge to inviscid models at a linear rate in energy norm.

Nonlinear acoustic models include quadratic nonlinearities of specific forms.

Numerical experiments confirm theoretical convergence results.

Abstract

We analyze the behavior of third-order in time linear and nonlinear sound waves in thermally relaxing fluids and gases as the sound diffusivity vanishes. The nonlinear acoustic propagation is modeled by the Jordan--Moore--Gibson--Thompson equation both in its Westervelt and in its Kuznetsov-type forms, that is, including quadratic nonlinearities of the type $(u^2)_{tt}$ and $ (u_t^2 + |\nabla u|^2)_t$. As it turns out, sufficiently smooth solutions of these equations converge in the energy norm to the solutions of the corresponding inviscid models at a linear rate. Numerical experiments illustrate our theoretical findings.