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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 |
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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