Nonlinear acoustics: Blackstock-Crighton equations with a periodic forcing term

TL;DR

This paper analyzes the Blackstock-Crighton equations with periodic forcing, demonstrating the existence of time-periodic solutions in a viscous, heat-conducting fluid, highlighting the role of dissipation in preventing resonance.

Contribution

It provides a mathematical proof of the existence of periodic solutions for the Blackstock-Crighton equations with time-periodic forcing, using fixed-point methods and a priori estimates.

Findings

Existence of time-periodic solutions under certain conditions

Dissipative effects prevent resonance in the model

Solutions exist in three-dimensional bounded domains

Abstract

The Blackstock-Crighton equations describe the motion of a viscous, heat-conducting, compressible fluid. They are used as models for acoustic wave propagation in a medium in which both nonlinear and dissipative effects are taken into account. In this article, a mathematical analysis of the Blackstock-Crighton equations with a time-periodic forcing term is carried out. For arbitrary time-periodic data (sufficiently restricted in size) it is shown that a time-periodic solution of the same period always exists. This implies that the dissipative effects are sufficient to avoid resonance within the Blackstock-Crighton models. The equations are considered in a three-dimensional bounded domain with both non-homogeneous Dirichlet and Neumann boundary values. Existence of a solution is obtained via a fixed-point argument based on appropriate a priori estimates for the linearized equations.