Entropy solutions to the Buckley--Leverett equations
type: Item
instance of: scholarly article
title: Entropy solutions to the Buckley--Leverett equations
zbMATH Open document ID: 0949.76081
DOI: 10.1007/BF02674603
author: P. I. Plotnikov
author: Stephan Luckhaus
published in: Siberian Mathematical Journal
publication date: 2000-08-30T00:00:00Z
full work available at URL: https://eudml.org/doc/120656
review text: The authors consider a mathematical model of the flow of two immiscible fluids in Hele-Shaw cells of a porous medium. One has to determine a function \(s\) (\(s\) is the saturation of one of the fluids), pressure \(p\), and vector-function \(v(x,t)\) (the rate of penetration of the mixture) satisfying the equations \(v\cdot\nabla A(s)- b\cdot\nabla s= 0\), \(\text{div}(k(s)\nabla p)= f(s,v)\), \(v= -k(s)\nabla p\) in \(\Omega\), and boundary conditions \(v\cdot n|_{\partial\Omega}= 0\), \(s|_{\partial\Omega^+}= s_0(x)\). Here \(\Omega\) is a bounded domain in \(\mathbb{R}^2\) with smooth boundary, \(\partial\Omega^+= \{x\in \partial\Omega: b\cdot n> 0\}\) with \(n\) being a unit exterior normal to \(\partial\Omega\), \(b\) is a given smooth vector field, \(A\) and \(f\) are given functions, and \(k\) is a given positive function. By an entropy solution the authors mean functions \(v\in L_2(\Omega)\) and \(s\in L_\infty(\Omega)\) satisfying the relations \[ \int_\Omega((\Psi(s)\cdot v- \Phi(s)\cdot b)\cdot\nabla\eta- (\Phi(s)\text{ div }b+ \Psi(s) f)\eta) dx+ \int_{\partial\Omega^+} \Phi(s_0)\eta b\cdot n ds\geq 0, \] \[ \int_\Omega (\nabla p\nabla\xi\cdot k(s)- f\xi) dx= 0,\quad v= -k(s)\nabla p \] for all functions \(\xi,\eta: \Omega\to \mathbb{R}\) and \(\phi,\Phi,\Psi: \mathbb{R}\to \mathbb{R}\) such that \(\xi\in C^\infty(\overline\Omega)\), \(\eta\in C^\infty_0(\mathbb{R}^2)\), \(\eta\geq 0\), \(\text{spt }\eta\cap (\partial\Omega\setminus \partial\Omega^+)= \varnothing\), \(\varphi\in C^1(\mathbb{R})\), \(\varphi'\geq 0\), \(\Psi'= a(s)\varphi(s)\) \((a(s)= A'(s))\), and \(\Phi'= \varphi\). Under some natural conditions, the authors prove the existence of measure-valued entropy solutions. These solutions are limits of approximate solutions obtained by elliptic regularization of the problem. Under some additional conditions on the functions \(A\) and \(k\), it is shown that the solutions satisfy the above definition of an entropy solution on certain parts of th…
Mathematics Subject Classification ID: 76S05
Mathematics Subject Classification ID: 76D27
Mathematics Subject Classification ID: 76T99
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 1498469
zbMATH Keywords: two-phase flow
zbMATH Keywords: Hele-Shaw cells
zbMATH Keywords: porous medium
zbMATH Keywords: existence of measure-valued entropy solutions
zbMATH Keywords: elliptic regularization
MaRDI profile type: Publication
cites work: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid
cites work: Q4184562
cites work: FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES
cites work: Q3320728
cites work: Measure-valued solutions to conservation laws
cites work: Q5686382
cites work: Q4183803
OpenAlex ID: W1972672665
description: scientific article; zbMATH DE number 1498469