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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  March 2020 Early access  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 and 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.

[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.

[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.

[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.

[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).

[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.

[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.

[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. 

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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).

[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.

[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.

[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. 

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

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