Vanishing relaxation time limit of the Jordan--Moore--Gibson--Thompson wave equation with Neumann and absorbing boundary conditions

TL;DR

This paper investigates the Jordan--Moore--Gibson--Thompson wave equation with Neumann and absorbing boundary conditions, analyzing how solutions behave as the relaxation time parameter approaches zero, and connecting it to classical acoustics models.

Contribution

It establishes well-posedness and energy bounds for the JMGT equation that are independent of the relaxation parameter, enabling the derivation of the Westervelt equation as a singular limit.

Findings

Solutions remain well-posed as relaxation time approaches zero.

Energy bounds can be made independent of the relaxation parameter.

The Westervelt equation is recovered as a limit of the JMGT equation.

Abstract

We study the Jordan--Moore--Gibson--Thompson (JMGT) equation, a third order in time wave equation that models nonlinear sound propagation, in the practically relevant setting of Neumann and absorbing boundary conditions. In the analysis, we pay special attention to dependencies on the coefficient $\tau$ of the third order time derivative that plays the physical role of relaxation time. We establish local in time well-posedness and derive energy bounds that can be made independent of $\tau$ under appropriate conditions. This fact allows us to pass to the limit $\tau\to0$ and recover solutions of a classical model in nonlinear acoustics, the Westervelt equation, as singular limits of solutions to the JMGT equation.