# American Institute of Mathematical Sciences

September  2015, 14(5): 2021-2042. doi: 10.3934/cpaa.2015.14.2021

## Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line

 1 School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China 2 School of Mathematics, South China University of Technology, Guangzhou 510641, China

Received  September 2014 Revised  January 2015 Published  June 2015

In this paper we study an asymptotic behavior of a solution to the initial boundary value problem for a viscous liquid-gas two-phase flow model in a half line $R_+:=(0,\infty).$ Our idea mainly comes from [23] and [29] which describe an isothermal Navier-Stokes equation in a half line. We obtain the convergence rate of the time global solution towards corresponding stationary solution in Eulerian coordinates. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. These theorems are proved by a weighted energy method.
Citation: 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 and Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021
##### References:
 [1] M. Baudin, C. Berthon, F. Coquel, R. Masson and Q. H. Tran, A relaxation method for two-phase flow models with hydrodynamic closure law, Numer. Math., 99 (2005), 411-440. doi: 10.1007/s00211-004-0558-1. [2] M. Baudin, F. Coquel and Q. H. Tran, A semi-implicit relaxation scheme for modeling two-phase flow in a pipeline, SIAM J. Sci. Comput., 27 (2005), 914-936. doi: 10.1137/030601624. [3] C. E. Brennen, Fundamentals of Multiphase Flow, Cambridge University Press, New York, 2005. [4] C. H. Chang and M. S. Lion, A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and $AUSM^{+}$-up scheme, J. Comput. Phys., 225 (2007), 840-873. doi: 10.1016/j.jcp.2007.01.007. [5] J. Cortes, A. Debussche and I. Toumi, A density perturbation method to study the eigenstructure of two-phase flow equation systems, J. Comput. Phys., 147 (1998), 463-484. doi: 10.1006/jcph.1998.6096. [6] J. M. Delhaye, M. Giot and M. L. Riethmuller, Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering, Von Karman Institute, McGraw-Hill, New York, 1981. [7] 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. [8] S. Evje and K. K. Fjelde, On a rough AUSM scheme for a one dimensional two-phase model, Comput. Fluids, 32 (2003), 1497-1530. doi: 10.1016/S0045-7930(02)00113-5. [9] S. Evje and T. Flåtten, On the wave structure of two-phase flow models, SIAM J. Appl. Math., 67 (2006), 487-511. doi: 10.1137/050633482. [10] 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. [11] 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. [12] 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. [13] S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells, SIAM J. Math. Anal., 43 (2011), 1887-1922. doi: 10.1137/100813932. [14] 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. [15] S. Evje, Q. Q. Liu and C. J. Zhu, Asymptotic stability of the compressible gas-liquid model with well-formation interaction and gravity, J. Differential Equations, 257 (2014), 3226-3271. doi: 10.1016/j.jde.2014.06.012. [16] L. Fan, Q. Q. Liu and C. J. Zhu, Convergence rates to stationary solutions of a gas-liquid model with external forces, Nonlinearity, 27 (2012), 2875-2901. doi: 10.1088/0951-7715/25/10/2875. [17] T. Flåtten and S. T. Munkejord, The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model, ESAIM: Math. Mod. Num. Anal., 40 (2006), 735-764. doi: 10.1051/m2an:2006032. [18] H. A. Friis and S. Evje, Asymptotic behavior of a compressible two-phase model with well-formation interaction, J. Differential Equations, 254 (2013), 3957-3993. doi: 10.1016/j.jde.2013.02.001. [19] 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. [20] M. Ishii and T. Hibiki, Thermo-fluid Dynamics of Two-Phase Flow, Springer-Verlag, New York, 2006. doi: 10.1007/978-0-387-29187-1. [21] Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space, Comm. Math. Phys., 266 (2006), 401-430. doi: 10.1007/s00220-006-0017-1. [22] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 0624.76095. [23] S. Kawashima, S. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys., 240 (2003), 483-500. doi: 10.1007/s00220-003-0909-2. [24] N. I. Kolev, Multiphase Flow Dynamics, Vol. 1. Fundamentals, Springer-Verlag, Berlin, 2005. [25] N. I. Kolev, Multiphase Flow Dynamics, Vol. 2. Thermal and Mechanical Interactions, Springer-Verlag, Berlin, 2005. [26] 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. [27] R. J. Lorentzen and K. K. Fjelde, Use of slopelimiter techniques in traditional numerical methods for multi-phase flow in pipelines and wells, Int. J. Numer. Meth. Fluids, 48 (2005), 723-745. [28] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 0543.76099. [29] T. Nakamura, S. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations, 241 (2007), 94-111. doi: 10.1016/j.jde.2007.06.016. [30] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107-132. [31] A. Prosperetti and G. Tryggvason (Editors), Computational Methods for Multiphase Flow, Cambridge University Press, New York, 2007. [32] 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. [33] L. Yao, T. Zhang and C. J. Zhu, Existence of 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. [34] 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. [35] 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. [36] 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. [37] Y. H. Zhang and C. J. Zhu, Global existence and optimal convergence rate for the strong solutions in $H^{2}$ to the 3D viscous liquid-gas two-phase flow model,, preprint., ().

show all references

##### References:
 [1] M. Baudin, C. Berthon, F. Coquel, R. Masson and Q. H. Tran, A relaxation method for two-phase flow models with hydrodynamic closure law, Numer. Math., 99 (2005), 411-440. doi: 10.1007/s00211-004-0558-1. [2] M. Baudin, F. Coquel and Q. H. Tran, A semi-implicit relaxation scheme for modeling two-phase flow in a pipeline, SIAM J. Sci. Comput., 27 (2005), 914-936. doi: 10.1137/030601624. [3] C. E. Brennen, Fundamentals of Multiphase Flow, Cambridge University Press, New York, 2005. [4] C. H. Chang and M. S. Lion, A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and $AUSM^{+}$-up scheme, J. Comput. Phys., 225 (2007), 840-873. doi: 10.1016/j.jcp.2007.01.007. [5] J. Cortes, A. Debussche and I. Toumi, A density perturbation method to study the eigenstructure of two-phase flow equation systems, J. Comput. Phys., 147 (1998), 463-484. doi: 10.1006/jcph.1998.6096. [6] J. M. Delhaye, M. Giot and M. L. Riethmuller, Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering, Von Karman Institute, McGraw-Hill, New York, 1981. [7] 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. [8] S. Evje and K. K. Fjelde, On a rough AUSM scheme for a one dimensional two-phase model, Comput. Fluids, 32 (2003), 1497-1530. doi: 10.1016/S0045-7930(02)00113-5. [9] S. Evje and T. Flåtten, On the wave structure of two-phase flow models, SIAM J. Appl. Math., 67 (2006), 487-511. doi: 10.1137/050633482. [10] 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. [11] 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. [12] 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. [13] S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells, SIAM J. Math. Anal., 43 (2011), 1887-1922. doi: 10.1137/100813932. [14] 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. [15] S. Evje, Q. Q. Liu and C. J. Zhu, Asymptotic stability of the compressible gas-liquid model with well-formation interaction and gravity, J. Differential Equations, 257 (2014), 3226-3271. doi: 10.1016/j.jde.2014.06.012. [16] L. Fan, Q. Q. Liu and C. J. Zhu, Convergence rates to stationary solutions of a gas-liquid model with external forces, Nonlinearity, 27 (2012), 2875-2901. doi: 10.1088/0951-7715/25/10/2875. [17] T. Flåtten and S. T. Munkejord, The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model, ESAIM: Math. Mod. Num. Anal., 40 (2006), 735-764. doi: 10.1051/m2an:2006032. [18] H. A. Friis and S. Evje, Asymptotic behavior of a compressible two-phase model with well-formation interaction, J. Differential Equations, 254 (2013), 3957-3993. doi: 10.1016/j.jde.2013.02.001. [19] 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. [20] M. Ishii and T. Hibiki, Thermo-fluid Dynamics of Two-Phase Flow, Springer-Verlag, New York, 2006. doi: 10.1007/978-0-387-29187-1. [21] Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space, Comm. Math. Phys., 266 (2006), 401-430. doi: 10.1007/s00220-006-0017-1. [22] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 0624.76095. [23] S. Kawashima, S. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys., 240 (2003), 483-500. doi: 10.1007/s00220-003-0909-2. [24] N. I. Kolev, Multiphase Flow Dynamics, Vol. 1. Fundamentals, Springer-Verlag, Berlin, 2005. [25] N. I. Kolev, Multiphase Flow Dynamics, Vol. 2. Thermal and Mechanical Interactions, Springer-Verlag, Berlin, 2005. [26] 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. [27] R. J. Lorentzen and K. K. Fjelde, Use of slopelimiter techniques in traditional numerical methods for multi-phase flow in pipelines and wells, Int. J. Numer. Meth. Fluids, 48 (2005), 723-745. [28] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. doi: 0543.76099. [29] T. Nakamura, S. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations, 241 (2007), 94-111. doi: 10.1016/j.jde.2007.06.016. [30] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107-132. [31] A. Prosperetti and G. Tryggvason (Editors), Computational Methods for Multiphase Flow, Cambridge University Press, New York, 2007. [32] 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. [33] L. Yao, T. Zhang and C. J. Zhu, Existence of 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. [34] 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. [35] 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. [36] 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. [37] Y. H. Zhang and C. J. Zhu, Global existence and optimal convergence rate for the strong solutions in $H^{2}$ to the 3D viscous liquid-gas two-phase flow model,, preprint., ().
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