Entropy solutions to the Buckley--Leverett equations
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 the domain \(\Omega\).
existence of measure-valued entropy solutions
scientific article; zbMATH DE number 1498469