## 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)

### MSC:

 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