A diffusion system for fluid in fractured media. (English) Zbl 0753.35045

The authors consider an abstract diffusion system which stems from a physical problem associated with the flow of fluid through fractured porous rock: find \({\mathbf u}:=(u,v)\) such that \({d\over dt}{\mathbf A}({\mathbf u})+{\mathbf B}({\mathbf u})\ni{\mathbf f}\). Here \({\mathbf A}({\mathbf u})=(A_ 1(u),A_ 2(v))\) and \(A_ 1,A_ 2\) are maximal monotone graphs in \(\mathbb{R}\times\mathbb{R}\), \({\mathbf B}({\mathbf u})=(\gamma(u-v),-\gamma(u-v)+B(v))\), \(\gamma=\gamma(x,\cdot)\) is a monotone function on \(\mathbb{R}\) (for any \(x\) from a bounded domain \(\Omega\subset\mathbb{R}^ n\)), and \(B\) is some nonlinear (possibly degenerate) second order elliptic operator, \({\mathbf f}=(f,g)\) are data. Moreover the Dirichlet condition upon \(v\) on \(\partial\Omega\) and an initial condition are imposed.
Assuming certain growth conditions on \(A_ 1,A_ 2,\dots\), semigroup theory in \(L^ 1\) is applied to obtain existence and uniqueness of generalized solutions for the problem which arises if \(a={\mathbf A}({\mathbf u})\) is introduced as the new unknown: the Crandall Liggett theorem is used to show that the sequence of solutions of a suited difference scheme converges to \(a\) in \(C([0,T],L^ 1(\Omega))\). Moreover, the authors show that nonnegative data yield nonnegative solutions \(a\), and deduce some bounds for the pressures \(u,v\).
Reviewer: K.J.Witsch (Essen)


35K65 Degenerate parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
76S05 Flows in porous media; filtration; seepage
74R99 Fracture and damage
35A35 Theoretical approximation in context of PDEs
35J60 Nonlinear elliptic equations
47H20 Semigroups of nonlinear operators