Global existence for a two-phase flow model with cross-diffusion

Esther S. Daus; Josipa-Pina Milišić; Nicola Zamponi

doi:10.3934/dcdsb.2019198

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2020, Volume 25, Issue 3: 957-979. Doi: 10.3934/dcdsb.2019198

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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

^* Corresponding author: Esther S. Daus
^* Corresponding author: Esther S. Daus

Received: December 31, 2018

Revised: April 30, 2019

Early access: September 2019

Published: March 2020

The first and the third author acknowledge partial support from the Austrian Science Fund (FWF), grants P22108, P24304, W1245, P27352 and P30000. All three authors were partially supported by the bilaterial project No. HR 04/2018 of the Austrian Exchange Sevice OeAD together with the Ministry of Science and Education of the Republic of Croatia MZO

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K65, 35K70, 35K55, 76S05; Secondary: 35Q35.

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