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

Part of: Two-phase and multiphase flows Hyperbolic equations and systems 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
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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References

1Brennen, C. E.. Fundamentals of multiphase flow (New York: Cambridge University Press, 2005).CrossRefGoogle Scholar
2Cui, H. B., Wang, W. J., Yao, L. and Zhu, C. J.. Decay rates of a nonconservative compressible generic two-fluid model. SIAM J. Math. Anal. 48 (2016), 470–512.CrossRefGoogle Scholar
3Danchin, R.. Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math. 141 (2000), 579–614.CrossRefGoogle Scholar
4Danchin, R.. Global existence in critical spaces for flows of compressible viscous and heat-conductive gases. Arch. Rational Mech. Anal. 16 (2001), 1–39.CrossRefGoogle Scholar
5Danchin, R.. Fourier analysis methods for PDEs, in: Lecture notes, 1 November 2005.Google Scholar
6Duan, R. J. and Ma, H. F.. Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity. Indiana Univ. Math. J. 5 (2008), 2299–2319.CrossRefGoogle Scholar
7Evans, L. C.. Partial Differential Equations (Providence, RI: Amer Math Soc, 1998).Google Scholar
8Evje, S.. Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells. SIAM J. Appl. Math. 43 (2011), 1887–1922.CrossRefGoogle Scholar
9Evje, S.. Global weak solutions for a compressible gas-liquid model with well-formation interaction. J. Differ. Equ. 251 (2011), 2352–2386.CrossRefGoogle Scholar
10Evje, S. and Flåtten, T.. Hybrid flux-splitting schemes for a common two-fluid model. J. Comput. Phys. 192 (2003), 175–210.CrossRefGoogle Scholar
11Evje, S. and Flåtten, T.. On the wave structure of two-phase flow models. SIAM J. Appl. Math. 67 (2006), 487–511.CrossRefGoogle Scholar
12Evje, S. and Karlsen, K. H.. Global existence of weak solutions for a viscous two-phase model. J. Differ. Equ. 245 (2008), 2660–2703.CrossRefGoogle Scholar
13Evje, S. and Karlsen, K. H.. Global weak solutions for a viscous liquid-gas model with singular pressure law. Commun. Pure Appl. Anal. 8 (2009), 1867–1894.CrossRefGoogle Scholar
14Evje, S., Flåtten, T. and Friis, H. A.. Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum. Nonlinear Anal. 70 (2009), 3864–3886.CrossRefGoogle Scholar
15Evje, S., Wang, W. J. and Wen, H. Y.. Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model. Arch. Rational Mech. Anal. 221 (2016), 2352–2386.CrossRefGoogle Scholar
16Evje, S., Wen, H. Y. and Zhu, C. J.. On global solutions to the viscous liquid-gas model with unconstrained transition to single-phase flow. Math. Model. Meth. Appl. Sci. 27 (2017), 323–346.CrossRefGoogle Scholar
17Fan, L., Liu, Q. Q. and Zhu, C. J.. Convergence rates to stationary solutions of a gas-liquid model with external forces. Nonlinearity 27 (2012), 2875–2901.CrossRefGoogle Scholar
18Friis, H. A., Evje, S. and Flåtten, T.. A numerical study of characteristic slow-transient behavior of a compressible 2D gas-liquid two-fluid model. Adv. Appl. Math. Mech. 1 (2009), 166–200.Google Scholar
19Guo, Z. H., Yang, J. and Yao, L.. Global strong solution for a three-dimensional viscous liquid-gas two-phase flow model with vacuum. J. Math. Phys. 52 (2011), 093102.CrossRefGoogle Scholar
20Hao, C. C. and Li, H. L.. Well-posedness for a multidimensional viscous liquid-gas two-phase flow model. SIAM J. Math. Anal. 44 (2012), 1304–1332.CrossRefGoogle Scholar
21Hou, X. F. and Wen, H. Y.. A blow-up criterion of strong solutions to a viscous liquid-gas two-phase flow model with vacuum in 3D. Nonlinear Anal. 75 (2012), 5229–5237.CrossRefGoogle Scholar
22Ishii, M.. Thermo-fluid dynamic theory of two-phase flow (Paris: Eyrolles, 1975).Google Scholar
23Kobayashi, T.. Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in ℝ3. J. Differ. Equ. 184 (2002), 587–619.CrossRefGoogle Scholar
24Kobayashi, T. and Shibata, Y.. Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in ℝ3. Comm. Math. Phys. 200 (1999), 621–659.CrossRefGoogle Scholar
25Li, H. L. and Zhang, T.. Large time behavior of isentropic compressible Navier-Stokes system in ℝ3. Math. Methods Appl. Sci. 34 (2011), 670–682.CrossRefGoogle Scholar
26Liu, T. P. and Zeng, Y.. Compressible Navier-Stokes equations with zero heat conductivity. J. Differ. Equ. 153 (1999), 225–291.CrossRefGoogle Scholar
27Liu, Q. Q. and Zhu, C. J.. Asymptotic behavior of a viscous liquid-gas model with mass-dependent viscosity and vacuum. J. Differ. Equ. 252 (2012), 2492–2519.CrossRefGoogle Scholar
28Matsumura, A. and Nishida, T.. The initial value problem for the equations of motion of compressible viscous and heat conductive fluids. Proc. Japan Acad. Ser. A 55 (1979), 337–342.CrossRefGoogle Scholar
29Matsumura, A. and Nishida, T.. The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20 (1980), 67–104.CrossRefGoogle Scholar
30Ruan, L. Z.. Smoothing effect on the damping mechanism for an inviscid two-phase gas-liquid model. P. Roy. Soc. Edinb. A, 144A (2014), 351–362.CrossRefGoogle Scholar
31Tan, Z. and Wang, Y. J.. On hyperbolic-dissipative systems of composite type. J. Differ. Equ. 260 (2016), 1091–1125.CrossRefGoogle Scholar
32Wang, W. J. and Wang, W. K.. Large time behavior for the system of a viscous liquid-gas two-phase flow model in ℝ3. J. Differ. Equ. 261 (2016), 5561–5589.CrossRefGoogle Scholar
33Wen, H. Y., Yao, L. and Zhu, C. J.. A blow-up criterion of strong solution to a 3D viscous liquid-gas two-phase flow model with vacuum. J. Math. Pures Appl. 97 (2012), 204–229.CrossRefGoogle Scholar
34Wu, G. C. and Zhang, Y. H.. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete Cont. Dyn. Sys.-B 23 (2018), 1411–1429.Google Scholar
35Wu, G. C., Tan, Z. and Huang, J.. Global existence and large time behavior for the system of compressible adiabatic flow through porous media in ℝ3. J. Differ. Equ. 255 (2013), 865–880.CrossRefGoogle Scholar
36Yao, L. and Zhu, C. J.. Free boundary value problem for a viscous two-phase model with mass-dependent viscosity. J. Differ. Equ. 247 (2009), 2705–2739.CrossRefGoogle Scholar
37Yao, L. and Zhu, C. J.. Existence and uniqueness of global weak solution to a two-phase flow model with vacuum. Math. Ann. 349 (2011), 903–928.CrossRefGoogle Scholar
38Yao, L., Zhang, T. and Zhu, C. J.. Existence and asymptotic behavior of global weak solutions to a 2D viscous liquid-gas two-phase flow model. SIAM J. Math. Anal. 42 (2010), 1874–1897.CrossRefGoogle Scholar
39Yao, L., Zhang, T. and Zhu, C. J.. A blow-up criterion for a 2D viscous liquid-gas two-phase flow model. J. Differ. Equ. 250 (2011), 3362–3378.CrossRefGoogle Scholar
40Yao, L., Zhu, C. J. and Zi, R. Z.. Incompressible limit of viscous liquid-gas two-phase flow model. SIAM J. Math. Anal. 44 (2012), 3324–3345.CrossRefGoogle Scholar
41Yao, L., Yang, J. and Guo, Z. H.. Blow-up criterion for 3D viscous liquid-gas two-phase flow model. J. Math. Anal. Appl. 395 (2012), 175–190.CrossRefGoogle Scholar
42Zhang, Y. H.. Decay of the 3D viscous liquid-gas two-phase flow model with damping. J. Math. Phys. 57 (2016), 081517.CrossRefGoogle Scholar
43Zhang, Y. H.. Decay of the 3D inviscid liquid-gas two-phase flow model. Z. Angew. Math. Phys. 67 (2016), 54.CrossRefGoogle Scholar
44Zhang, Y. H. and Zhu, C. J.. Global existence and optimal convergence rates for the strong solutions in H 2 to the 3D viscous liquid-gas two-phase flow model. J. Differ. Equ. 258 (2015), 2315–2338.CrossRefGoogle Scholar
45Ziemer, W.. Weakly differentiable functions (Berlin: Springer, 1989).CrossRefGoogle Scholar