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Global well-posedness and large time behaviour of the viscous liquid-gas two-phase flow model in ℝ3

Part of: Hyperbolic equations and systems Two-phase and multiphase flows Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  14 March 2019

Guochun Wu  and
Yinghui Zhang
Guochun Wu
Affiliation:
Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou362021, China (guochunwu@126.com)
Yinghui Zhang*
Affiliation:
School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi541004, P.R. China (zhangyinghui0910@126.com)
*
*Corresponding author.
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Abstract

We investigate the Cauchy problem of the viscous liquid-gas two-phase flow model in ℝ3. Under the assumption that the initial data is close to the constant equilibrium state in the framework of Sobolev space H2(ℝ3), the Cauchy problem is shown to be globally well-posed by an energy method. If additionally, for 1 ⩽ p < 6/5, Lp-norm of the initial perturbation is bounded, the optimal convergence rates of the solutions in Lq-norm with 2 ⩽ q ⩽ 6 and optimal convergence rates of their spatial derivatives in L2-norm are also obtained by combining spectral analysis with energy methods.

Keywords

Two-phase flow modelglobal existenceoptimal convergence rates

MSC classification

Primary: 76T10: Liquid-gas two-phase flows, bubbly flows
Secondary: 35Q35: PDEs in connection with fluid mechanics 35L65: Conservation laws

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1999 - 2024
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Copyright
Copyright © Royal Society of Edinburgh 2019

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