Nonlinear acoustic equations of fractional higher order at the singular limit

Abstract

When high-frequency sound waves travel through media with anomalous diffusion, such as biological tissues, their motion can be described by nonlinear acoustic equations of fractional higher order. 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 a singular limit analysis 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 in nonlinear acoustics. We furthermore establish the convergence rates and thus determine the error one makes when exchanging local and nonlocal models. The analysis rests upon the uniform bounds for the solutions of the acoustic equations with fractional higher-order derivatives, obtained through a testing procedure tailored to the coercivity property of the involved (weakly) singular memory kernel.