Nonlinear acoustic equations of fractional higher order at the singular limit

TL;DR

This paper investigates nonlinear fractional higher-order acoustic wave equations in media with anomalous diffusion, analyzing their relation to classical equations through singular limit analysis as relaxation times approach zero.

Contribution

It establishes the connection between fractional higher-order models and classical acoustic equations via singular limit analysis, providing a rigorous justification for their approximation in nonlinear acoustics.

Findings

Models approximate Westervelt, Blackstock, or Kuznetsov equations depending on conditions.

Uniform bounds for solutions with fractional derivatives are obtained.

Singular limit analysis clarifies the relation to classical wave equations.

Abstract

When high-frequency sound waves travel through media with anomalous diffusion, such as biological tissues, their motion can be described by nonlinear wave equations of fractional higher order. These can be understood as nonlocal generalizations of the Jordan-Moore-Gibson-Thompson equations in nonlinear acoustics. In this work, we relate them to the classical second-order acoustic equations and, in this sense, justify them as their approximations for small relaxation times. To this end, we perform the singular limit analysis for a class of corresponding nonlocal wave models and determine their behavior as the relaxation time tends to zero. We show that, depending on the nonlinearities and assumptions on the data, these models can be seen as approximations of the Westervelt, Blackstock, or Kuznetsov wave equations. The analysis rests upon the uniform bounds for the solutions of the equations with fractional higher-order derivatives, obtained through a testing procedure tailored to the coercivity property of the involved (weakly) singular memory kernels.