Abstract

In this talk we will study a model describing the evolution of the thickness of a grounded shallow ice sheet. The evolution of the thickness of the grounded shallow ice sheet under consideration is modelled via Glen’s power law. Moreover, since the ice sheet thickness is constrained to be nonnegative, the problem under consideration is an obstacle problem.

The first part of the talk is devoted to the recovery of the model describing this phenomenon. A rigorous modelling exercise shows that the obtained model is time-dependent, and is governed by a set of variational inequalities that involve nonlinearities in the time derivative and in the elliptic term.

The second part of the talks is devoted to establishing the existence of solutions for the formal model recovered beforehand. In order to establish the existence of solutions for the time-dependent model we recovered, formally, upon completion of the aforementioned modelling exercise, we first depart from a penalized relaxation, and we show - by resorting to a discretization in time - that the corresponding relaxed problem admits at least one solution. Secondly, by means of Dubinskii's lemma and other new results and new inequalities, we extract compactness for the family of solutions of the relaxed problems and we show that this family of solutions converges to a solution of a doubly nonlinear parabolic variational inequality akin to the one that was recovered formally.

These results have been obtained in collaboration with Roger Temam (Indiana University Bloomington)