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.
Citation: |
[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.![]() ![]() ![]() |