The Kuznetsov and Blackstock equations of nonlinear acoustics with nonlocal-in-time dissipation

TL;DR

This paper investigates nonlocal nonlinear acoustic wave equations with fractional dissipation, analyzing well-posedness, limiting behavior, and convergence rates, with applications to heat flux laws in ultrasonics.

Contribution

It extends the analysis of Kuznetsov and Blackstock equations to include fractional nonlocal dissipation and studies their limiting behavior with respect to a small parameter.

Findings

Established well-posedness under minimal assumptions on memory kernels.

Analyzed the limiting behavior as the dissipation parameter tends to zero.

Derived convergence rates of solutions in the energy norm.

Abstract

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin--Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag-Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm.