Resumen: Recent studies on the mechanism governing the Laurentide ice sheet oscillations of the Last Ice Age focus on the most critical effect of the basal hydraulic processes enhanced when the ice is sliding along soft deformable beds. To understand the import of this, we consider Fowler and Johnson's 0-D hydrological flow model describing the sudden and rapid movements forward (surges)
of a till-based 1-D ice sheet sliding on a hat soft bed. The basic idea is that the interplay between the ice sheets dynamics, the basal drainage system, and the sliding law can generate a surging behaviour. Mathematically this means that a multiple valued relationship between the ice flux and the ice thickness arises and the mass conservation equation turns out to be of multivalued type for some special values of the dimensionless parameters involved in the model. Assuming that a multiple valued ice flux law of the Fowler and Johnson type herds, we prove the existence of a weak bounded discontinuous solution to the system which becomes periodic after a suitable time.
Palabras clave: system of nonlinear equations; ice sheet models; surges
Resumen: The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale
river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin.