Existence and stability for mathematical models of sedimentation-consolidation processes in several space dimensions

In this paper, we study several spatially multidimensional initial-boundary value problems modelling sedimentation-consolidation processes of a flocculated suspension. This solid-fluid mixture is considered as two superimposed continuous media differing in density and viscosity. The phenomenological foundation and derivation of the mathematical model are based on the previous work by R. Bürger et al. (2000, Z. Angew. Math. Mech. 80, 79-92). We study the full coupling of the conservation of mass equation and the conservation of linear momentum equation. For different types of regularization, we establish energy estimates. The dissipative nature of the whole system assures the existence as well as the stability (long time asymptotics) of a solution of the system (provided that the viscosities of the fluid are large enough). Moreover, the energy estimates might serve as the foundation for the design of numerical algorithms to simulate the system.