Asymptotic behaviors for Blackstock's model of thermoviscous flow

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 $n⩾5$ 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.

Indiana Univ. Math. J. (in press)

References in corpus (1)

• The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case

Cited by in corpus (2)

• Optimal large-time estimates and singular limits for thermoelastic plate equations with the Fourier law
• Large-time asymptotic behaviors for linear Blackstock's model of thermoviscous flow