Mathematical Analysis of Diffusion Models in Poro-Elastic Media

We consider a coupled system of mixed hyperbolic-parabolic type which describes the Biot consolidation model in poro-elasticity as well as a coupled quasi-static problem in thermoelasticity. The pioneering work of M. A. Biot [1, 2, 3, 4] has given rise to numerous extensions. Among them, R. E. Showalter [7] developed an existence, uniqueness and regularity theory for the degenerate quasi-static system using the theory of linear degenerate evolution equations in Hilbert space while C.M. Dafermos studied the classical coupled thermo-elasticity system which describes the flow of heat through an elastic structure [5]. Herein, we choose to consider the full dynamic coupled system which is a more general model including the previous ones and which describes a more complete phenomenon of consolidation. As far as applications are concerned, Biot’s theory is generally involved for the study of a soil under load but it also provides a good theory for the ultrasonic propagation in fluid-saturated porous media like cancellous bone. Our objective here is to develop the existence-uniqueness theory for the one dimensional systems both in the linear and nonlinear cases using classical functional arguments in the Sobolev background. In the linear case, we present a fixed point method based on J. L. Lions’ results [6] to establish the existence of a solution to the problem while in the nonlinear one, our approach involves Galerkin approximations coupled with a monotonicity method.