On a degenerate parabolic/hyperbolic system in glaciology giving rise to a free boundary

Abstract

The authors present and study a problem which models the evolution of the ice sheet in the Laurentide. They consider a one-dimensional problem in (3-dimensional) space which involves three parameters: the ice thickness h , the amount of water flux Q and the accumulated ice velocity ξ . Considering the mass conservation law, the momentum or balance equations, and introducing the special glaciology relations already described in the specialized literature, they write the coupled system involving these three unknowns. After some computations, they are led to some coupled system of parabolic and hyperbolic nonlinear and possibly degenerate equations. Initial and boundary conditions are introduced which correspond to the special case of the Hudson region. Replacing the data by piecewise constant approximations with respect to the time variable, the authors then present some stationary discretized coupled system for which they define the notion of weak solution. The purpose of this work is to obtain some existence result for this discretized system. This is done using an iterative scheme which decouples the three equations. An existence result is proved for each of these decoupled equations. The first equation (for the discretization of h ) is studied using the notion of super- and subsolution and comparison principles. The second equation (for the discretization of Q ) involves a maximal monotone graph and is studied, first by replacing this multivalued graph by some single-valued maximal monotone graph and then by passing to the limit. The study of the last equation is very easy. Then uniform estimates are established on these approximate solutions. This leads to an asymptotic result which finally proves the existence result for the original discretized system. The last part is devoted to the qualitative study of the function Q , which must be nonnegative. The authors prove the existence of a boundary layer, corresponding to the boundary of the region {Q>0} . The present work justifies earlier observations made by the second author when solving the problem numerically