Global existence and estimates for Blackstock's model of thermoviscous flow with second sound phenomena

According to the Lighthill scheme of approximation procedures, we establish the Blackstock-Cattaneo model in this paper to describe nonlinear acoustics' propagations with second sound phenomena in perfect gases under the irrotational flow, which is an approximated higher-order equation for Navi...

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Bibliographic Details
Published in:Journal of Differential Equations 2022-07, Vol.324, p.76-101
Main Authors: Liu, Yan, Qin, Xulong, Zhang, Shuanghu
Format: Article
Language:English
Subjects:
Cauchy problem
Decay estimates
Global existence of Sobolev solution
Nonlinear acoustics
The Cattaneo law
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Summary:According to the Lighthill scheme of approximation procedures, we establish the Blackstock-Cattaneo model in this paper to describe nonlinear acoustics' propagations with second sound phenomena in perfect gases under the irrotational flow, which is an approximated higher-order equation for Navier-Stokes-Cattaneo equations. This paper is devoted to the mathematical analysis of the linear and nonlinear higher-order evolution equations in Rn. Concerning the linearized Cauchy problem for the Blackstock-Cattaneo equation, by means of the Fourier analysis and the WKB analysis, we will derive sharp L2 estimates of solutions with additional L1 regularity for initial data. For another, we demonstrate global (in time) well-posedness for the nonlinear Blackstock-Cattaneo equation with small initial data and suitable regularity of Sobolev solutions.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2022.04.001