American Institute of Mathematical Sciences

Advanced
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

Haiyan Yin 1, and  Changjiang Zhu 2,

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.
Keywords: initial boundary value problem, convergence rate, Viscous liquid-gas two-phase flow model, stationary solution, weighted energy method..
Mathematics Subject Classification: Primary: 35B35, 35B40; Secondary: 76T1.
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 & 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,, \emph{Numer. Math.}, 99 (2005), 411. doi: 10.1007/s00211-004-0558-1. Google Scholar

[2]

M. Baudin, F. Coquel and Q. H. Tran, A semi-implicit relaxation scheme for modeling two-phase flow in a pipeline,, \emph{SIAM J. Sci. Comput.}, 27 (2005), 914. doi: 10.1137/030601624. Google Scholar

[3]

C. E. Brennen, Fundamentals of Multiphase Flow,, Cambridge University Press, (2005). Google Scholar

[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,, \emph{J. Comput. Phys.}, 225 (2007), 840. doi: 10.1016/j.jcp.2007.01.007. Google Scholar

[5]

J. Cortes, A. Debussche and I. Toumi, A density perturbation method to study the eigenstructure of two-phase flow equation systems,, \emph{J. Comput. Phys.}, 147 (1998), 463. doi: 10.1006/jcph.1998.6096. Google Scholar

[6]

J. M. Delhaye, M. Giot and M. L. Riethmuller, Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering,, Von Karman Institute, (1981). Google Scholar

[7]

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

[8]

S. Evje and K. K. Fjelde, On a rough AUSM scheme for a one dimensional two-phase model,, \emph{Comput. Fluids}, 32 (2003), 1497. doi: 10.1016/S0045-7930(02)00113-5. Google Scholar

[9]

S. Evje and T. Flåtten, On the wave structure of two-phase flow models,, \emph{SIAM J. Appl. Math.}, 67 (2006), 487. doi: 10.1137/050633482. Google Scholar

[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,, \emph{Nonlinear Anal.}, 70 (2009), 3864. doi: 10.1016/j.na.2008.07.043. Google Scholar

[11]

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

[12]

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

[13]

S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells,, \emph{SIAM J. Math. Anal.}, 43 (2011), 1887. doi: 10.1137/100813932. Google Scholar

[14]

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

[15]

S. Evje, Q. Q. Liu and C. J. Zhu, Asymptotic stability of the compressible gas-liquid model with well-formation interaction and gravity,, \emph{J. Differential Equations}, 257 (2014), 3226. doi: 10.1016/j.jde.2014.06.012. Google Scholar

[16]

L. Fan, Q. Q. Liu and C. J. Zhu, Convergence rates to stationary solutions of a gas-liquid model with external forces,, \emph{Nonlinearity}, 27 (2012), 2875. doi: 10.1088/0951-7715/25/10/2875. Google Scholar

[17]

T. Flåtten and S. T. Munkejord, The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model,, \emph{ESAIM: Math. Mod. Num. Anal.}, 40 (2006), 735. doi: 10.1051/m2an:2006032. Google Scholar

[18]

H. A. Friis and S. Evje, Asymptotic behavior of a compressible two-phase model with well-formation interaction,, \emph{J. Differential Equations}, 254 (2013), 3957. doi: 10.1016/j.jde.2013.02.001. Google Scholar

[19]

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

[20]

M. Ishii and T. Hibiki, Thermo-fluid Dynamics of Two-Phase Flow,, Springer-Verlag, (2006). doi: 10.1007/978-0-387-29187-1. Google Scholar

[21]

Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space,, \emph{Comm. Math. Phys.}, 266 (2006), 401. doi: 10.1007/s00220-006-0017-1. Google Scholar

[22]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion,, \emph{Comm. Math. Phys.}, 101 (1985), 97. doi: 0624.76095. Google Scholar

[23]

S. Kawashima, S. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space,, \emph{Comm. Math. Phys.}, 240 (2003), 483. doi: 10.1007/s00220-003-0909-2. Google Scholar

[24]

N. I. Kolev, Multiphase Flow Dynamics,, Vol. 1. Fundamentals, (2005). Google Scholar

[25]

N. I. Kolev, Multiphase Flow Dynamics,, Vol. 2. Thermal and Mechanical Interactions, (2005). Google Scholar

[26]

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

[27]

R. J. Lorentzen and K. K. Fjelde, Use of slopelimiter techniques in traditional numerical methods for multi-phase flow in pipelines and wells,, \emph{Int. J. Numer. Meth. Fluids}, 48 (2005), 723. Google Scholar

[28]

A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids,, \emph{Comm. Math. Phys.}, 89 (1983), 445. doi: 0543.76099. Google Scholar

[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,, \emph{J. Differential Equations}, 241 (2007), 94. doi: 10.1016/j.jde.2007.06.016. Google Scholar

[30]

M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws,, \emph{Funkcial. Ekvac.}, 41 (1998), 107. Google Scholar

[31]

A. Prosperetti and G. Tryggvason (Editors), Computational Methods for Multiphase Flow,, Cambridge University Press, (2007). Google Scholar

[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,, \emph{J. Math. Pures Appl.}, 97 (2012), 204. doi: 10.1016/j.matpur.2011.09.005. Google Scholar

[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,, \emph{SIAM J. Math. Anal.}, 42 (2010), 1874. doi: 10.1137/100785302. Google Scholar

[34]

L. Yao, T. Zhang and C. J. Zhu, A blow-up criterion for a 2D viscous liquid-gas two-phase flow model,, \emph{J. Differential Equations}, 250 (2011), 3362. doi: 10.1016/j.jde.2010.12.006. Google Scholar

[35]

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

[36]

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

[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., (). Google Scholar

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,, \emph{Numer. Math.}, 99 (2005), 411. doi: 10.1007/s00211-004-0558-1. Google Scholar

[2]

M. Baudin, F. Coquel and Q. H. Tran, A semi-implicit relaxation scheme for modeling two-phase flow in a pipeline,, \emph{SIAM J. Sci. Comput.}, 27 (2005), 914. doi: 10.1137/030601624. Google Scholar

[3]

C. E. Brennen, Fundamentals of Multiphase Flow,, Cambridge University Press, (2005). Google Scholar

[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,, \emph{J. Comput. Phys.}, 225 (2007), 840. doi: 10.1016/j.jcp.2007.01.007. Google Scholar

[5]

J. Cortes, A. Debussche and I. Toumi, A density perturbation method to study the eigenstructure of two-phase flow equation systems,, \emph{J. Comput. Phys.}, 147 (1998), 463. doi: 10.1006/jcph.1998.6096. Google Scholar

[6]

J. M. Delhaye, M. Giot and M. L. Riethmuller, Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering,, Von Karman Institute, (1981). Google Scholar

[7]

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

[8]

S. Evje and K. K. Fjelde, On a rough AUSM scheme for a one dimensional two-phase model,, \emph{Comput. Fluids}, 32 (2003), 1497. doi: 10.1016/S0045-7930(02)00113-5. Google Scholar

[9]

S. Evje and T. Flåtten, On the wave structure of two-phase flow models,, \emph{SIAM J. Appl. Math.}, 67 (2006), 487. doi: 10.1137/050633482. Google Scholar

[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,, \emph{Nonlinear Anal.}, 70 (2009), 3864. doi: 10.1016/j.na.2008.07.043. Google Scholar

[11]

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

[12]

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

[13]

S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells,, \emph{SIAM J. Math. Anal.}, 43 (2011), 1887. doi: 10.1137/100813932. Google Scholar

[14]

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

[15]

S. Evje, Q. Q. Liu and C. J. Zhu, Asymptotic stability of the compressible gas-liquid model with well-formation interaction and gravity,, \emph{J. Differential Equations}, 257 (2014), 3226. doi: 10.1016/j.jde.2014.06.012. Google Scholar

[16]

L. Fan, Q. Q. Liu and C. J. Zhu, Convergence rates to stationary solutions of a gas-liquid model with external forces,, \emph{Nonlinearity}, 27 (2012), 2875. doi: 10.1088/0951-7715/25/10/2875. Google Scholar

[17]

T. Flåtten and S. T. Munkejord, The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model,, \emph{ESAIM: Math. Mod. Num. Anal.}, 40 (2006), 735. doi: 10.1051/m2an:2006032. Google Scholar

[18]

H. A. Friis and S. Evje, Asymptotic behavior of a compressible two-phase model with well-formation interaction,, \emph{J. Differential Equations}, 254 (2013), 3957. doi: 10.1016/j.jde.2013.02.001. Google Scholar

[19]

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

[20]

M. Ishii and T. Hibiki, Thermo-fluid Dynamics of Two-Phase Flow,, Springer-Verlag, (2006). doi: 10.1007/978-0-387-29187-1. Google Scholar

[21]

Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space,, \emph{Comm. Math. Phys.}, 266 (2006), 401. doi: 10.1007/s00220-006-0017-1. Google Scholar

[22]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion,, \emph{Comm. Math. Phys.}, 101 (1985), 97. doi: 0624.76095. Google Scholar

[23]

S. Kawashima, S. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space,, \emph{Comm. Math. Phys.}, 240 (2003), 483. doi: 10.1007/s00220-003-0909-2. Google Scholar

[24]

N. I. Kolev, Multiphase Flow Dynamics,, Vol. 1. Fundamentals, (2005). Google Scholar

[25]

N. I. Kolev, Multiphase Flow Dynamics,, Vol. 2. Thermal and Mechanical Interactions, (2005). Google Scholar

[26]

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

[27]

R. J. Lorentzen and K. K. Fjelde, Use of slopelimiter techniques in traditional numerical methods for multi-phase flow in pipelines and wells,, \emph{Int. J. Numer. Meth. Fluids}, 48 (2005), 723. Google Scholar

[28]

A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids,, \emph{Comm. Math. Phys.}, 89 (1983), 445. doi: 0543.76099. Google Scholar

[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,, \emph{J. Differential Equations}, 241 (2007), 94. doi: 10.1016/j.jde.2007.06.016. Google Scholar

[30]

M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws,, \emph{Funkcial. Ekvac.}, 41 (1998), 107. Google Scholar

[31]

A. Prosperetti and G. Tryggvason (Editors), Computational Methods for Multiphase Flow,, Cambridge University Press, (2007). Google Scholar

[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,, \emph{J. Math. Pures Appl.}, 97 (2012), 204. doi: 10.1016/j.matpur.2011.09.005. Google Scholar

[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,, \emph{SIAM J. Math. Anal.}, 42 (2010), 1874. doi: 10.1137/100785302. Google Scholar

[34]

L. Yao, T. Zhang and C. J. Zhu, A blow-up criterion for a 2D viscous liquid-gas two-phase flow model,, \emph{J. Differential Equations}, 250 (2011), 3362. doi: 10.1016/j.jde.2010.12.006. Google Scholar

[35]

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

[36]

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

[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., (). Google Scholar

[1]

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
[2]

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
[3]

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
[4]

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
[5]

Steinar Evje, Kenneth Hvistendahl Karlsen. Global weak solutions for a viscous liquid-gas model with singular pressure law. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1867-1894. doi: 10.3934/cpaa.2009.8.1867
[6]

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
[7]

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
[8]

Dongfen Bian. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1591-1611. doi: 10.3934/dcdss.2016065
[9]

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
[10]

Yuning Liu, Wei Wang. On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115
[11]

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
[12]

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
[13]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019198
[14]

Feng Ma, Mingfang Ni. A two-phase method for multidimensional number partitioning problem. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 203-206. doi: 10.3934/naco.2013.3.203
[15]

Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917
[16]

Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015
[17]

Yangyang Qiao, Huanyao Wen, Steinar Evje. Compressible and viscous two-phase flow in porous media based on mixture theory formulation. Networks & Heterogeneous Media, 2019, 14 (3) : 489-536. doi: 10.3934/nhm.2019020
[18]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[19]

Mauro Garavello. Boundary value problem for a phase transition model. Networks & Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89
[20]

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

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences