Vanishing relaxation time dynamics of the Jordan Moore-Gibson-Thompson equation arising in nonlinear acoustics

TL;DR

This paper analyzes the vanishing relaxation time limit of the Jordan Moore-Gibson-Thompson equation in nonlinear acoustics, proving strong convergence to the Westervelt equation and providing convergence rates.

Contribution

It establishes strong convergence of solutions as relaxation time approaches zero, addressing an open question and extending previous weak convergence results with explicit rates.

Findings

Solutions converge strongly to the Westervelt equation as relaxation time vanishes.

The paper provides explicit rates of convergence for solutions with higher regularity.

Convergence results hold on an infinite time horizon, improving upon prior weak convergence findings.

Abstract

The (third-order in time) JMGT equation \cite{Jordan2,HCP} is a nonlinear (quasi-linear) Partial Differential Equation (PDE) model introduced to describe a nonlinear propagation of sound in an acoustic medium. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second order in time equation referred to as Westervelt equation. Replacing Fourier's law by Maxwell-Cattaneo's law gives rise to the third order in time derivative scaled by a small parameter $\tau >0$, the latter represents the thermal relaxation time parameter and is intrinsic to the medium where the dynamics occur. In this paper we provide an asymptotic analysis of the third order model when $\tau \rightarrow 0 $. It is shown that the corresponding solutions converge {\it in a strong topology of the phase space } to a limit which is the solution of Westervelt equation. In addition, rate of convergence is provided for solutions displaying higher order regularity. This addresses an open question raised in \cite{kaltev2}, where a related JMGT equation has been studied and {\it weak star } convergence of the solutions when $\tau \rightarrow 0$ has been established. Thus, our main contribution is showing {\it strong convergence on infinite time horizon,} along with related rates of convergence valid on a finite time horizon. The key to unlocking the difficulty owns to a tight control and propagation of the "smallness" of the initial data in carrying the estimates at three different topological levels. The rate of convergence allows one then to estimate the relaxation time needed for the signal to reach the target. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences.