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Diffusion of fluid in a fissured medium with microstructure. (English) Zbl 0764.35053

The authors continue their work on diffusion in fissured media [Differ. Integral Equ. 3, No. 2, 219-236 (1990; Zbl 0753.35045), J. Math. Anal. Appl. 155, No. 1, 1-20 (1991; Zbl 0711.60078)]. They consider the flow of a fluid through a fissured medium. This is assumed to be a structure of porous and permeable cells which are separated from each other by a system of fissures. The majority of fluid transport will occur along flow paths through the fissure system. There is assumed to be no direct flow between adjacent cells, since they are individually isolated by the fissures, but the dynamics of the flux exchange between each cell and its surrounding fissures is a major aspect in the investigation. Thus, in such a context, we have (nonlinear parabolic) equations prescribing the flow on the global scale of the fissure system, equations giving the flow on the microscale of the individual cell at a specific point in the fissure system, and constraints prescribing the transfer of fluid between the cells and surrounding medium.
The authors begin by stating and resolving the stationary forms of the systems in a variational formulation by monotone operators from Banach spaces to their duals. The authors develop an abstract Green’s theorem to describe the resolution as the sum of a partial differential equation and a complementary boundary operator. Conditions of coercitivity type assert the existence of generalized solutions of the variational equations. The stationary operator in the original (parabolic) equations is shown to be accretive in the space, and so one obtains a generalized solution in the sense of the nonlinear semigroup theory for general Banach spaces. Hence, the initial boundary value problems are well posed in appropriate spaces.

MSC:

35K65 Degenerate parabolic equations
76S05 Flows in porous media; filtration; seepage
35D05 Existence of generalized solutions of PDE (MSC2000)
35K55 Nonlinear parabolic equations
35A15 Variational methods applied to PDEs
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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