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The global attractor for a class of extensible beams with nonlocal weak damping
Global existence for a two-phase flow model with cross-diffusion
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 |
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.
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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