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.
Reviewer: Arkadi Berezovski (Tallinn)
MSC:
| 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 |
Keywords:
porous linear elastic medium;coupled quasi-static problems;slightly compressible viscous fluid;equilibrium equation for momentum;diffusion equation;Darcy flow;regularity;evolution equations;Hilbert space;semigroup theory;differential operators;initial-boundary value problem;well-posed parabolic problem;existence;uniqueness;weak solutionCite
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