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

**Title: **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.