Continuity of the saturation in the flow of two immiscible fluids in a porous medium
type: Item
instance of: scholarly article
title: Continuity of the saturation in the flow of two immiscible fluids in a porous medium
zbMATH Open document ID: 1246.35110
author: Emmanuele DiBenedetto
author: Ugo Gianazza
author: Vincenzo Vespri
published in: Indiana University Mathematics Journal
publication date: 2012-04-10T00:00:00Z
full work available at URL: http://www.iumj.indiana.edu/IUMJ/ABS/2010/4004
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 6022807
zbMATH Keywords: degenerate elliptic parabolic systems
zbMATH Keywords: flow of two immiscible fluids
zbMATH Keywords: Kruzkov-Sukorjanski transformation
zbMATH Keywords: Buckley-Leverett coupling
MaRDI profile type: Publication
OpenAlex ID: W1985281038
DOI: 10.1512/IUMJ.2010.59.4004
review text: The authors study the weakly coupled system NEWLINE\[NEWLINE \begin{aligned} & v_t - \text{div }[A(v)\nabla v + \vec B(v)] = \vec V\cdot \nabla C(v) \quad \text{in } E_T = E\times (0,T],\\ &\text{div }\vec V = 0, \end{aligned} NEWLINE\]NEWLINE where \(E \subset \mathbb R^N\) is a bounded domain with boundary \(\partial E\) of class \(C^1\). This system is separated into a parabolic equation for \(v\), and an elliptic equation for \(u\), and it arises in the theory of flow of immiscible fluids in a porous medium. The unknown functions \(u\) and \(v\) and the equations above represent the pressure and the saturation respectively due to the Kruzkov-Sukorjanski transformation, subject to Darcy's law and the Buckley-Leverett coupling.NEWLINENEWLINEUnder suitable structural assumptions on \(A\), \(\vec B\) and \(C\), the authors prove that if \((u,v)\) satisfy the above-mentioned system locally in the weak sense, then the saturation \(v\) is a locally continuous function in \(E_T\), irrespective of the nature of the degeneracy of the principal part of the system. Moreover, for every compact subset \(K_T \subset E_T\), there exists a continuous function \(\omega(\cdot)\) that can be determined a priori only in terms of the data and the distance from \(K_T\) to the parabolic boundary of \(E_T\), such that \(\omega(0) = 0\), and NEWLINE\[NEWLINE |v(x_1,t_1) - v(x_2,t_2)| \leq \omega(|x_1 - x_2| + |t_1 - t_2|^{1/2}) NEWLINE\]NEWLINE for \((x_i,t_i) \in K_T\).
reviewed by: Vladimir N. Grebenev
description: scientific article; zbMATH DE number 6022807