# American Institute of Mathematical Sciences

• Previous Article
Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model
• DCDS-B Home
• This Issue
• Next Article
A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model
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. Evje, T. 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. Fan, Q. 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. Friis, S. 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. Wen, L. 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. Yao, T. 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. Yao, T. 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. Yao, C. 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. Yao, J. 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. Evje, T. 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. Fan, Q. 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. Friis, S. 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. Wen, L. 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. Yao, T. 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. Yao, T. 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. Yao, C. 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. Yao, J. 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.
 [1] Theodore Tachim Medjo. A two-phase flow model with delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137 [2] Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure & Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011 [3] Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1 [4] T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665 [5] Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006 [6] Ming Mei, Bruno Rubino, Rosella Sampalmieri. Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain. Kinetic & Related Models, 2012, 5 (3) : 537-550. doi: 10.3934/krm.2012.5.537 [7] Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541 [8] Feimin Huang, Dehua Wang, Difan Yuan. Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3535-3575. doi: 10.3934/dcds.2019146 [9] Yasuhito Miyamoto. Global bifurcation and stable two-phase separation for a phase field model in a disk. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 791-806. doi: 10.3934/dcds.2011.30.791 [10] K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591 [11] Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks & Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401 [12] Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 395-430. doi: 10.3934/dcds.2019016 [13] Steinar Evje, Huanyao Wen, Lei Yao. Global solutions to a one-dimensional non-conservative two-phase model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1927-1955. doi: 10.3934/dcds.2016.36.1927 [14] Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1217-1251. doi: 10.3934/dcdsb.2013.18.1217 [15] Jingrui Su. Global existence and low Mach number limit to a 3D compressible micropolar fluids model in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3423-3434. doi: 10.3934/dcds.2017145 [16] Theodore Tachim Medjo. Pullback $\mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2141-2169. doi: 10.3934/dcds.2018088 [17] Haiyan Yin, Changjiang Zhu. Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021 [18] Yingshan Chen, Mei Zhang. A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions. Kinetic & Related Models, 2016, 9 (3) : 429-441. doi: 10.3934/krm.2016001 [19] Haibo Cui, Qunyi Bie, Zheng-An Yao. Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1395-1410. doi: 10.3934/dcdsb.2018156 [20] Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. On the $\Gamma$-limit for a non-uniformly bounded sequence of two-phase metric functionals. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 411-426. doi: 10.3934/dcds.2015.35.411

2017 Impact Factor: 0.972