| Contents | In this talk, which is partly based on results previously obtained by the speaker and Roger Temam, we present a thermodynamical model governing the evolution of the surface elevation of a grounded shallow ice sheet coupled with the evolution of the internal temperature of one such ice sheet.
The governing model for the evolution of the ice surface elevation is degenerate, nonlinear, and such that admissible ice surface elevations must be greater than or equal to the bedrock elevation, which is a given and sufficiently smooth function. The thermal model we are considering, which is a simplification of the original one, is posed over a moving domain, determined by the ice surface elevation, and contains a highly nonlinear source term. Besides, admissible temperatures are required to satisfy physically motivated constraints. Together, the constraint that the ice surface elevation lie above the bedrock and the physical constraints on the temperature turn this coupled system into an obstacle problem.
After establishing a model that complies with the physics of ice sheets, we aim to prove the existence of weak solutions to one such model. To this end, we re-cast the problem so that the temperature evolution is posed on a fixed domain, and apply a change of variables that removes the degeneracy in the ice surface elevation model, transforming the original parabolic problem into a doubly nonlinear parabolic one. To prove existence, we start from a relaxed version suggested by the physics and show that, as the relaxation parameter tends to zero, the approximate solutions converge — in a suitable sense — to solutions of the sharp-limit model, which takes the form of coupled evolutionary variational inequalities. |
