Analysis of Cross-Diffusion Systems for Fluid Mixtures Driven by a Pressure Gradient

The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy's law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field.

[1]  Nicola Zamponi,et al.  Existence analysis of a single-phase flow mixture model with van der Waals pressure , 2016, 1612.04161.

[2]  Ansgar Jüngel,et al.  Existence Analysis of a Single-Phase Flow Mixture with van der Waals Pressure , 2018, SIAM J. Math. Anal..

[3]  Wolfgang Dreyer,et al.  A mixture theory of electrolytes containing solvation effects , 2013 .

[4]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[5]  Ansgar Jüngel,et al.  The boundedness-by-entropy method for cross-diffusion systems , 2015 .

[6]  Danielle Hilhorst,et al.  A NONLINEAR PARABOLIC-HYPERBOLIC SYSTEM FOR CONTACT INHIBITION OF CELL-GROWTH , 2012 .

[7]  Ansgar Jüngel,et al.  Entropy Methods for Diffusive Partial Differential Equations , 2016 .

[8]  V. Girault,et al.  Vector potentials in three-dimensional non-smooth domains , 1998 .

[9]  Gonzalo Galiano,et al.  Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem , 2014 .

[10]  Yuan-hui Li,et al.  Equation of state of water and sea water , 1967 .

[11]  D. Duffy.  Second‐Order Parabolic Differential Equations , 2013 .

[12]  Martin Burger,et al.  Sorting Phenomena in a Mathematical Model For Two Mutually Attracting/Repelling Species , 2017, SIAM J. Math. Anal..

[13]  E. Feireisl,et al.  Singular Limits in Thermodynamics of Viscous Fluids , 2009 .

[14]  Esther S. Daus,et al.  Rigorous mean-field limit and cross-diffusion , 2018, Zeitschrift für angewandte Mathematik und Physik.

[15]  M. Ughi,et al.  On a multidimensional model for the codiffusion of isotopes: localization and asymptotic behavior , 2016 .

[16]  Fernando A. Morales,et al.  A Darcy–Brinkman model of fractures in porous media , 2016, 1611.05318.

[17]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[18]  Rüdiger Müller,et al.  Bulk-Surface Electrothermodynamics and Applications to Electrochemistry , 2018, Entropy.

[19]  M E Gurtin,et al.  On interacting populations that disperse to avoid crowding: preservation of segregation , 1985, Journal of mathematical biology.

[20]  Alfio Quarteroni,et al.  A multiscale Darcy–Brinkman model for fluid flow in fractured porous media , 2011, Numerische Mathematik.

[21]  Dieter Bothe,et al.  Continuum thermodynamics of chemically reacting fluid mixtures , 2013, 1401.5991.

[22]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[23]  Rüdiger Müller,et al.  Overcoming the shortcomings of the Nernst-Planck model. , 2013, Physical chemistry chemical physics : PCCP.

[24]  Daniel Z. Zanger.  The Inhomogeneous Neumann Problem in Lipschitz Domains , 2000 .

[25]  A. Esposito,et al.  Nonlinear degenerate cross-diffusion systems with nonlocal interaction , 2017, 1710.01653.

[26]  Piotr Gwiazda,et al.  A two-species hyperbolic–parabolic model of tissue growth , 2018, Communications in Partial Differential Equations.

[27]  M. Burger,et al.  Segregation and Gap Formation in Cross-Diffusion Models , 2019, 1906.03712.

[28]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[29]  Harald Garcke,et al.  A multiphase Cahn--Hilliard--Darcy model for tumour growth with necrosis , 2017, 1701.06656.

[30]  Gonzalo Galiano,et al.  On a cross-diffusion segregation problem arising from a model of interacting particles , 2013, 1311.3276.

[31]  Morton E. Gurtin,et al.  A note on interacting populations that disperse to avoid crowding , 1984 .

[32]  Filippo Santambrogio,et al.  Splitting Schemes and Segregation in Reaction Cross-Diffusion Systems , 2017, SIAM J. Math. Anal..