Diffusion in poro-elastic media. (English) Zbl 0979.74018

The author considers a system of equations which models diffusion in porous linear elastic medium saturated by a slightly compressible viscous fluid. The system consists of the equilibrium equation for momentum and diffusion equation for Darcy flow. The existence-uniqueness-regularity theory is developed by using the theory of evolution equations in Hilbert space and the classical semigroup theory. The author gives a description of appropriate Lebesgue and Sobolev spaces, and constructs differential operators that represent coupled elasticity and diffusion equations. It is proved that the quasi-static initial-boundary value problem is a well-posed parabolic problem, and the corresponding strong solution is sufficiently regular. The corresponding estimates are obtained directly from the abstract theory, which allows to prove the existence and uniqueness of a weak solution under weak assumptions on the data. A special interest is given to the problem when the evolution is parabolic or merely hyperbolic.


74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76S05 Flows in porous media; filtration; seepage
35Q72 Other PDE from mechanics (MSC2000)
35Q35 PDEs in connection with fluid mechanics
76R50 Diffusion
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