Large-time asymptotic behaviors for linear Blackstock's model of thermoviscous flow

TL;DR

This paper analyzes the long-time behavior of solutions to Blackstock's model for thermoviscous acoustic waves, deriving detailed asymptotic profiles that improve previous estimates and include second-order corrections.

Contribution

It provides the first- and second-order asymptotic profiles for the linear Blackstock's model without Becker's assumption, using refined analytical techniques.

Findings

Derived optimal first- and second-order asymptotic profiles for solutions.

Improved estimates over previous work for lower-dimensional cases.

Identified the leading term and second-order profiles with weighted $L^1$ data.

Abstract

In the classical theory of acoustic waves, Blackstock's model was proposed in 1963 to characterize the propagation of sound in thermoviscous fluids. In this paper, we investigate large-time asymptotic behaviors of the linear Cauchy problem for general Blackstock's model (that is, without Becker's assumption on monatomic perfect gases). We derive first- and second-order asymptotic profiles of solution as $t\gg1$ by applying refined WKB analysis and Fourier analysis. Our results not only improve optimal estimates in [Chen-Ikehata-Palmieri, \emph{Indiana Univ. Math. J.} (2023)] for lower dimensional cases, but also illustrate the optimal leading term and novel second-order profiles of solution with additional weighted $L^1$ data.