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June 2018, 23(4): 1411-1429. doi: 10.3934/dcdsb.2018157

Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain

1. 

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

2. 

Department of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China

* Corresponding author: Yinghui Zhang

Received  November 2016 Revised  March 2018 Published  April 2018

Fund Project: The first author is supported by National Natural Science Foundation of China #11701193, #11671086. The second author is supported by Hunan Provincial Natural Science Foundation of China #2017JJ2105, and National Natural Science Foundation of China #11571280, #11771150, #11301172, 11226170, and National Scholarship Fund in Hunan province cooperation projects

In this paper, we investigate global existence and asymptotic behavior of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain with no-slip boundary. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in $H^2(Ω)$. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

Citation: Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157
References:
[1]

R. A. Adams, Sobolev Space, Academic Press, New York, 1975.

[2]

C. E. Brennen, Fundamentals of Multiphase Flow, Cambridge University Press, New York, 2005. doi: 10.1017/CBO9780511807169.

[3]

R. J. Duan and H. F. Ma, Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity, Indiana Univ. Math. J., 5 (2008), 2299-2319. doi: 10.1512/iumj.2008.57.3326.

[4]

S. Evje and T. Flåtten, Hybrid flux-splitting schemes for a common two-fluid model, J. Comput. Phys., 192 (2003), 175-210. doi: 10.1016/j.jcp.2003.07.001.

[5]

S. EvjeT. Flåtten and H. A. Friis, Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum, Nonlinear Anal., 70 (2009), 3864-3886. doi: 10.1016/j.na.2008.07.043.

[6]

S. Evje and K. H. Karlsen, Global existence of weak solutions for a viscous two-phase model, J. Differential Equations, 245 (2008), 2660-2703. doi: 10.1016/j.jde.2007.10.032.

[7]

S. Evje and K. H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, Commun. Pure Appl. Anal., 8 (2009), 1867-1894. doi: 10.3934/cpaa.2009.8.1867.

[8]

S. Evje and T. Flåtten, On the wave structure of two-phase flow models, SIAM J. Appl. Math., 67 (2006/07), 487-511.

[9]

S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells, SIAM J. Appl. Math., 43 (2011), 1887-1922. doi: 10.1137/100813932.

[10]

S. Evje, Global weak solutions for a compressible gas-liquid model with well-formation interaction, J. Differential Equations, 251 (2011), 2352-2386. doi: 10.1016/j.jde.2011.07.013.

[11]

L. FanQ. Q. Liu and C. J. Zhu, Convergence rates to stationary solutions of a gas-liquid model with external forces, Nonlinearity, 25 (2012), 2875-2901. doi: 10.1088/0951-7715/25/10/2875.

[12]

H. A. FriisS. Evje and T. Flåtten, 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.

[13]

Z. H. Guo, J. Yang and L. Yao, Global strong solution for a three-dimensional viscous liquid-gas two-phase flow model with vacuum, J. Math. Phys., 52 (2011), 093102, 14pp.

[14]

C. C. Hao and H. L. Li, Well-posedness for a multidimensional viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 44 (2012), 1304-1332. doi: 10.1137/110851602.

[15]

X. F. Hou and H. Y. Wen, 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. doi: 10.1016/j.na.2012.04.039.

[16]

M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris, 1975.

[17]

Y. Kagei and S. Kawashima, Local solvability of initial boundary value problem for a quasilinear hyperbolic-parabolic system, J. Hyperbolic Differ. Equ., 3 (2006), 195-232. doi: 10.1142/S0219891606000768.

[18]

Q. Q. Liu and C. J. Zhu, Asymptotic behavior of a viscous liquid-gas model with mass-dependent viscosity and vacuum, J. Differential Equations, 252 (2012), 2492-2519. doi: 10.1016/j.jde.2011.10.018.

[19]

T. P. Liu and Y. Zeng, Compressible Navier-Stokes equations with zero heat conductivity, J. Differential Equations, 153 (1999), 225-291. doi: 10.1006/jdeq.1998.3554.

[20]

A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Commun. Math. Phys., 89 (1983), 445-464.

[21]

H. Y. WenL. Yao and C. J. Zhu, 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. doi: 10.1016/j.matpur.2011.09.005.

[22]

L. Yao and C. J. Zhu, Free boundary value problem for a viscous two-phase model with mass-dependent viscosity, J. Differential Equations., 247 (2009), 2705-2739. doi: 10.1016/j.jde.2009.07.013.

[23]

L. Yao and C. J. Zhu, Existence and uniqueness of global weak solution to a two-phase flow model with vacuum, Math. Ann., 349 (2011), 903-928. doi: 10.1007/s00208-010-0544-0.

[24]

L. YaoT. Zhang and C. J. Zhu, 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. doi: 10.1137/100785302.

[25]

L. YaoT. Zhang and C. J. Zhu, A blow-up criterion for a 2D viscous liquid-gas two-phase flow model, J. Differential Equations, 250 (2011), 3362-3378. doi: 10.1016/j.jde.2010.12.006.

[26]

L. YaoC. J. Zhu and R. Z. Zi, Incompressible limit of viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 44 (2012), 3324-3345. doi: 10.1137/120862120.

[27]

L. YaoJ. Yang and Z. H. Guo, Blow-up criterion for 3D viscous liquid-gas two-phase flow model, J. Math. Anal. Appl., 395 (2012), 175-190. doi: 10.1016/j.jmaa.2012.05.018.

[28]

Y. H. Zhang and C. J. Zhu, Global existence and optimal convergence rates for the strong solutions in $H^2$ to the 3D viscous liquid-gas two-phase flow model, J. Differential Equations, 258 (2015), 2315-2338. doi: 10.1016/j.jde.2014.12.008.

[29]

W. Ziemer, Weakly Differentiable Functions, Springer, Berlin, 1989.

show all references

References:
[1]

R. A. Adams, Sobolev Space, Academic Press, New York, 1975.

[2]

C. E. Brennen, Fundamentals of Multiphase Flow, Cambridge University Press, New York, 2005. doi: 10.1017/CBO9780511807169.

[3]

R. J. Duan and H. F. Ma, Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity, Indiana Univ. Math. J., 5 (2008), 2299-2319. doi: 10.1512/iumj.2008.57.3326.

[4]

S. Evje and T. Flåtten, Hybrid flux-splitting schemes for a common two-fluid model, J. Comput. Phys., 192 (2003), 175-210. doi: 10.1016/j.jcp.2003.07.001.

[5]

S. EvjeT. Flåtten and H. A. Friis, Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum, Nonlinear Anal., 70 (2009), 3864-3886. doi: 10.1016/j.na.2008.07.043.

[6]

S. Evje and K. H. Karlsen, Global existence of weak solutions for a viscous two-phase model, J. Differential Equations, 245 (2008), 2660-2703. doi: 10.1016/j.jde.2007.10.032.

[7]

S. Evje and K. H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, Commun. Pure Appl. Anal., 8 (2009), 1867-1894. doi: 10.3934/cpaa.2009.8.1867.

[8]

S. Evje and T. Flåtten, On the wave structure of two-phase flow models, SIAM J. Appl. Math., 67 (2006/07), 487-511.

[9]

S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells, SIAM J. Appl. Math., 43 (2011), 1887-1922. doi: 10.1137/100813932.

[10]

S. Evje, Global weak solutions for a compressible gas-liquid model with well-formation interaction, J. Differential Equations, 251 (2011), 2352-2386. doi: 10.1016/j.jde.2011.07.013.

[11]

L. FanQ. Q. Liu and C. J. Zhu, Convergence rates to stationary solutions of a gas-liquid model with external forces, Nonlinearity, 25 (2012), 2875-2901. doi: 10.1088/0951-7715/25/10/2875.

[12]

H. A. FriisS. Evje and T. Flåtten, 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.

[13]

Z. H. Guo, J. Yang and L. Yao, Global strong solution for a three-dimensional viscous liquid-gas two-phase flow model with vacuum, J. Math. Phys., 52 (2011), 093102, 14pp.

[14]

C. C. Hao and H. L. Li, Well-posedness for a multidimensional viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 44 (2012), 1304-1332. doi: 10.1137/110851602.

[15]

X. F. Hou and H. Y. Wen, 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. doi: 10.1016/j.na.2012.04.039.

[16]

M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris, 1975.

[17]

Y. Kagei and S. Kawashima, Local solvability of initial boundary value problem for a quasilinear hyperbolic-parabolic system, J. Hyperbolic Differ. Equ., 3 (2006), 195-232. doi: 10.1142/S0219891606000768.

[18]

Q. Q. Liu and C. J. Zhu, Asymptotic behavior of a viscous liquid-gas model with mass-dependent viscosity and vacuum, J. Differential Equations, 252 (2012), 2492-2519. doi: 10.1016/j.jde.2011.10.018.

[19]

T. P. Liu and Y. Zeng, Compressible Navier-Stokes equations with zero heat conductivity, J. Differential Equations, 153 (1999), 225-291. doi: 10.1006/jdeq.1998.3554.

[20]

A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Commun. Math. Phys., 89 (1983), 445-464.

[21]

H. Y. WenL. Yao and C. J. Zhu, 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. doi: 10.1016/j.matpur.2011.09.005.

[22]

L. Yao and C. J. Zhu, Free boundary value problem for a viscous two-phase model with mass-dependent viscosity, J. Differential Equations., 247 (2009), 2705-2739. doi: 10.1016/j.jde.2009.07.013.

[23]

L. Yao and C. J. Zhu, Existence and uniqueness of global weak solution to a two-phase flow model with vacuum, Math. Ann., 349 (2011), 903-928. doi: 10.1007/s00208-010-0544-0.

[24]

L. YaoT. Zhang and C. J. Zhu, 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. doi: 10.1137/100785302.

[25]

L. YaoT. Zhang and C. J. Zhu, A blow-up criterion for a 2D viscous liquid-gas two-phase flow model, J. Differential Equations, 250 (2011), 3362-3378. doi: 10.1016/j.jde.2010.12.006.

[26]

L. YaoC. J. Zhu and R. Z. Zi, Incompressible limit of viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 44 (2012), 3324-3345. doi: 10.1137/120862120.

[27]

L. YaoJ. Yang and Z. H. Guo, Blow-up criterion for 3D viscous liquid-gas two-phase flow model, J. Math. Anal. Appl., 395 (2012), 175-190. doi: 10.1016/j.jmaa.2012.05.018.

[28]

Y. H. Zhang and C. J. Zhu, Global existence and optimal convergence rates for the strong solutions in $H^2$ to the 3D viscous liquid-gas two-phase flow model, J. Differential Equations, 258 (2015), 2315-2338. doi: 10.1016/j.jde.2014.12.008.

[29]

W. Ziemer, Weakly Differentiable Functions, Springer, Berlin, 1989.

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