Asymptotic behaviors for Blackstock's model of thermoviscous flow

TL;DR

This paper analyzes the asymptotic behavior of Blackstock's nonlinear acoustics model in a0a0 space, deriving decay estimates, studying singular limits, and proving global existence of small data solutions.

Contribution

It provides new $L^2$ decay estimates, investigates singular limits, and establishes global solutions for the nonlinear Blackstock's model without Becker's assumption.

Findings

Optimal decay estimates for $n \u2265 5$

Growth rates for $n=3,4$

Existence of global small data solutions

Abstract

We study a fundamental model in nonlinear acoustics, precisely, the general Blackstock's model (that is, without Becker's assumption) in the whole space $\mathbb{R}^n$. This model describes nonlinear acoustics in perfect gases under the irrotational flow. By means of the Fourier analysis we will derive $L^2$ estimates for the solution of the linear homogeneous problem and its derivatives. Then, we will apply these estimates to study three different topics: the optimality of the decay estimates in the case $n\geqslant 5$ and the optimal growth rate for the $L^2$-norm of the solution for $n=3,4$; the singular limit problem in determining the first- and second-order profiles for the solution of the linear Blackstock's model with respect to the small thermal diffusivity; the proof of the existence of global (in time) small data Sobolev solutions with suitable regularity for a nonlinear Blackstock's model.