Partially saturated flow in a poroelastic medium
R.E. Showalter Ning Su
Discrete & Continuous Dynamical Systems - B 2001, 1(4): 403-420 doi: 10.3934/dcdsb.2001.1.403
The formulation and existence theory is presented for a system modeling diffusion of a slightly compressible fluid through a partially saturated poroelastic medium. Nonlinear effects of density, saturation, porosity and permeability variations with pressure are included, and the seepage surface is determined by a variational inequality on the boundary.
keywords: Poroelasticity maximal monotone operator partially saturated deformable porous medium consolidation quasi-static elliptic-parabolic system.
Semilinear degenerate parabolic systems and distributed capacitance models
Brooke L. Hollingsworth R.E. Showalter
Discrete & Continuous Dynamical Systems - A 1995, 1(1): 59-76 doi: 10.3934/dcds.1995.1.59
A two-scale microstructure model of current flow in a medium with continuously distributed capacitance is extended to include nonlinearities in the conductance across the interface between the local capacitors and the global conducting medium. The resulting degenerate system of partial differential equations is shown to be in the form of a semilinear parabolic evolution equation in Hilbert space. It is shown directly that such an equation is equivalent to a subgradient flow and, hence, displays the appropriate parabolic regularizing effects. Various limiting cases are identified and the corresponding convergence results obtained by letting selected parameters tend to infinity.
keywords: semilinear parabolic evolution equation in Hilbert space. subgradient flow microstructure model degenerate system the layered medium equation

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