American Institute of Mathematical Sciences

Advanced
March  2020, 25(3): 957-979. doi: 10.3934/dcdsb.2019198

Global existence for a two-phase flow model with cross-diffusion

Esther S. Daus 1,, , Josipa-Pina Milišić 2, and  Nicola Zamponi 1,

1. 

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria
2. 

University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia

* Corresponding author: Esther S. Daus

Received  January 2019 Revised  May 2019 Published  September 2019

Fund Project: The first and the third author acknowledge partial support from the Austrian Science Fund (FWF), grants P22108, P24304, W1245, P27352 and P30000. All three authors were partially supported by the bilaterial project No. HR 04/2018 of the Austrian Exchange Sevice OeAD together with the Ministry of Science and Education of the Republic of Croatia MZO.

In this work we study a degenerate pseudo-parabolic system with cross diffusion describing the evolution of the densities of an unsaturated two-phase flow mixture with dynamic capillary pressure in porous medium with saturation-dependent relaxation parameter and hypocoercive diffusion operator modeling cross diffusion. The equations are derived in a thermodynamically correct way from mass conservation laws. Global-in-time existence of weak solutions to the system in a bounded domain with equilibrium boundary conditions is shown. The main tools of the analysis are an entropy inequality and a crucial apriori bound which allows for controlling the degeneracy.

Keywords: Cross-diffusion, dynamic capillary pressure, degenerate nonlinear parabolic system, entropy method, existence of solutions.
Mathematics Subject Classification: Primary: 35K65, 35K70, 35K55, 76S05; Secondary: 35Q35.
Citation: Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198
References:
[1]

G. I. Barenblatt, V. M. Entov and V. M. Ryzhik, Theory of Fluid Flows Through Natural Rocks, Nedra Publishing House Moscow, 1972. Reissued by Springer 1996. doi: 10.1007/978-94-015-7899-8.  Google Scholar

[2]

J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluver Academic Publisher, 1990. doi: 10.1007/978-94-009-1926-6.  Google Scholar

[3]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries, Nonlinearity, 25 (2012), 961-990.  doi: 10.1088/0951-7715/25/4/961.  Google Scholar

[4]

X. Q. Chen, A. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskii lemmas, Acta Appl. Math., 133 (2014), 33-43.  doi: 10.1007/s10440-013-9858-8.  Google Scholar

[5]

W. Dreyer, P.-E. Druet, P. Gajewski and C. Guhlke, Analysis of improved Nernst-Planck-Poisson models of compressible isothermal electrolytes. Part Ⅲ: Compactness and convergence, WIAS, Preprint, (2017). Google Scholar

[6]

X. Cao and I. S. Pop, Uniqueness of weak solutions for a pseudo-parabolic equation modeling two phase flow in porous media, Appl. Math. Lett., 46 (2015), 25-30.  doi: 10.1016/j.aml.2015.01.022.  Google Scholar

[7]

X. Cao and I. S. Pop, Degenerate two-phase flow model in porpus media including dynamic effects in the capillary pressure: Existence of a weak solution, J. Diff. Equ., 260 (2016), 2418-2456.  doi: 10.1016/j.jde.2015.10.008.  Google Scholar

[8]

W. G. Gray and S. M. Hassanizadeh, Macroscale continuum mechanics for multiphase porous-media flow including phases, interfaces, common lines and commpon points, Adv. Water Resources, 21 (1998), 261-281.   Google Scholar

[9]

S. M. Hassanizadeh and W. G. Gray, Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries, Adv. Water Resources, 13 (1990), 169-186.  doi: 10.1016/0309-1708(90)90040-B.  Google Scholar

[10]

S. M. Hassanizadeh and W. G. Gray, Thermodynamic basis of capillary pressure in porous media, Water Resour. Res., 29 (1993), 3389-3405.  doi: 10.1029/93WR01495.  Google Scholar

[11]

A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001.  doi: 10.1088/0951-7715/28/6/1963.  Google Scholar

[12]

A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations, Springer Briefs in Mathematics, Springer, 2016. doi: 10.1007/978-3-319-34219-1.  Google Scholar

[13]

A. Jüngel, J. Mikyška and N. Zamponi, Existence analysis of a single-phase flow mixture model with van der Waals pressure, SIAM J. Math. Anal., 50 (2018), 1367-1395.  doi: 10.1137/16M1107024.  Google Scholar

[14]

A. Mikelić, A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure, J. Diff. Equ., 248 (2010), 1561-1577.  doi: 10.1016/j.jde.2009.11.022.  Google Scholar

[15]

J.-P. Milišić, The unsaturated flow in porous media with dynamic capillary pressure, J. Diff. Equ., 264 (2018), 5629-5658.  doi: 10.1016/j.jde.2018.01.014.  Google Scholar

[16]

M. Ruzhansky and M. Sugimoto, On global inversion of homogeneous maps, Bull. Math. Sci., 5 (2015), 13-18.  doi: 10.1007/s13373-014-0059-1.  Google Scholar

[17]

E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

show all references

References:
[1]

G. I. Barenblatt, V. M. Entov and V. M. Ryzhik, Theory of Fluid Flows Through Natural Rocks, Nedra Publishing House Moscow, 1972. Reissued by Springer 1996. doi: 10.1007/978-94-015-7899-8.  Google Scholar

[2]

J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluver Academic Publisher, 1990. doi: 10.1007/978-94-009-1926-6.  Google Scholar

[3]

M. Burger, B. Schlake and M.-T. Wolfram, Nonlinear Poisson-Nernst-Planck equations for ion flux through confined geometries, Nonlinearity, 25 (2012), 961-990.  doi: 10.1088/0951-7715/25/4/961.  Google Scholar

[4]

X. Q. Chen, A. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskii lemmas, Acta Appl. Math., 133 (2014), 33-43.  doi: 10.1007/s10440-013-9858-8.  Google Scholar

[5]

W. Dreyer, P.-E. Druet, P. Gajewski and C. Guhlke, Analysis of improved Nernst-Planck-Poisson models of compressible isothermal electrolytes. Part Ⅲ: Compactness and convergence, WIAS, Preprint, (2017). Google Scholar

[6]

X. Cao and I. S. Pop, Uniqueness of weak solutions for a pseudo-parabolic equation modeling two phase flow in porous media, Appl. Math. Lett., 46 (2015), 25-30.  doi: 10.1016/j.aml.2015.01.022.  Google Scholar

[7]

X. Cao and I. S. Pop, Degenerate two-phase flow model in porpus media including dynamic effects in the capillary pressure: Existence of a weak solution, J. Diff. Equ., 260 (2016), 2418-2456.  doi: 10.1016/j.jde.2015.10.008.  Google Scholar

[8]

W. G. Gray and S. M. Hassanizadeh, Macroscale continuum mechanics for multiphase porous-media flow including phases, interfaces, common lines and commpon points, Adv. Water Resources, 21 (1998), 261-281.   Google Scholar

[9]

S. M. Hassanizadeh and W. G. Gray, Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries, Adv. Water Resources, 13 (1990), 169-186.  doi: 10.1016/0309-1708(90)90040-B.  Google Scholar

[10]

S. M. Hassanizadeh and W. G. Gray, Thermodynamic basis of capillary pressure in porous media, Water Resour. Res., 29 (1993), 3389-3405.  doi: 10.1029/93WR01495.  Google Scholar

[11]

A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001.  doi: 10.1088/0951-7715/28/6/1963.  Google Scholar

[12]

A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations, Springer Briefs in Mathematics, Springer, 2016. doi: 10.1007/978-3-319-34219-1.  Google Scholar

[13]

A. Jüngel, J. Mikyška and N. Zamponi, Existence analysis of a single-phase flow mixture model with van der Waals pressure, SIAM J. Math. Anal., 50 (2018), 1367-1395.  doi: 10.1137/16M1107024.  Google Scholar

[14]

A. Mikelić, A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure, J. Diff. Equ., 248 (2010), 1561-1577.  doi: 10.1016/j.jde.2009.11.022.  Google Scholar

[15]

J.-P. Milišić, The unsaturated flow in porous media with dynamic capillary pressure, J. Diff. Equ., 264 (2018), 5629-5658.  doi: 10.1016/j.jde.2018.01.014.  Google Scholar

[16]

M. Ruzhansky and M. Sugimoto, On global inversion of homogeneous maps, Bull. Math. Sci., 5 (2015), 13-18.  doi: 10.1007/s13373-014-0059-1.  Google Scholar

[17]

E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

[1]

Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185
[2]

Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193
[3]

Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic & Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033
[4]

Yanxia Wu, Yaping Wu. Existence of traveling waves with transition layers for some degenerate cross-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (3) : 911-934. doi: 10.3934/cpaa.2012.11.911
[5]

Lianzhang Bao, Wenjie Gao. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2813-2829. doi: 10.3934/dcdsb.2017152
[6]

Peng Feng, Zhengfang Zhou. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1145-1165. doi: 10.3934/cpaa.2007.6.1145
[7]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[8]

Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 719-730. doi: 10.3934/dcds.2004.10.719
[9]

Y. S. Choi, Roger Lui, Yoshio Yamada. Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1193-1200. doi: 10.3934/dcds.2003.9.1193
[10]

Salomé Martínez, Wei-Ming Ni. Periodic solutions for a 3x 3 competitive system with cross-diffusion. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 725-746. doi: 10.3934/dcds.2006.15.725
[11]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[12]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081
[13]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228
[14]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435
[15]

F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239
[16]

Yaping Wu, Qian Xu. The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 367-385. doi: 10.3934/dcds.2011.29.367
[17]

Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam, Tian Xiang. A PDE model of intraguild predation with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3653-3661. doi: 10.3934/dcdsb.2017145
[18]

Michael Winkler, Dariusz Wrzosek. Preface: Analysis of cross-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : i-i. doi: 10.3934/dcdss.20202i
[19]

Maxime Breden, Christian Kuehn, Cinzia Soresina. On the influence of cross-diffusion in pattern formation. Journal of Computational Dynamics, 2021, 8 (2) : 213-240. doi: 10.3934/jcd.2021010
[20]

Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (140)
  • HTML views (243)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences