Structural stability of the Brinkman{Forchheimer equations for ows in porous media with variable porosity

In this paper we are concerned with the structural stability (continuous dependence of solutions) of the Brinkman{Forchheimer model for flows in porous media. Payne and Straughan established structural stability for the case of constant porosity. Herein we extend these results to porous media of variable porosity. The main differences when the porosity is no longer constant but a time-independent field variable are: i) the velocity field is not divergence-free, ii) the term related to the shear viscous stresses does not reduce to the Laplacian operator but involves the full deviatoric strain-rate tensor, and iii) there is an additional term due to normal viscous stresses. Herein we establish continuous dependence of solutions with respect to the physical parameters entering the equations, namely, the shear and bulk viscosities and the coeffcients of the linear (Darcy) and quadratic (Forchheimer) terms for the interfacial drag. More specifically, we show continuous dependence in the weighted L2 norm, with the porosity being the weight. Finally, we provide upper and lower bounds for the rate of decay of the kinetic energy. According to them, the kinetic energy decays exponentially with time but the decay rate depends on both the minimum and maximum porosities of the medium.