Applied Mathematics Seminar-Existence analysis of a single-phase flow mixture with van der Waals pressure

Speaker:  Nicola Zamponi 

Date and Time:  March 8, 2017, Thursday,3:00 pm-4:00pm

Place:   Rome 771

Abstract: The transport of single-phase fluid mixtures in porous media is described by cross-diffusion equations for the chemical concentrations. The equations are obtained in a thermodynamic consistent way from mass balance, Darcy's law, and the van der Waals equation of state for mixtures. Including diffusive fluxes, the global-in-time existence of weak solutions in a bounded domain with equilibrium boundary conditions is proved, using the boundedness-by-entropy method. Based on the free energy inequality, the large-time convergence of the solution to the constant equilibrium concentration is shown. For the two-species model and specific diffusion matrices, an integral inequality is proved, which reveals a maximum and minimum principle for the ratio of the concentrations. Without diffusive fluxes, the two-dimensional pressure is shown to converge exponentially fast to a constant. Numerical examples in one space dimension illustrate this convergence.