Abstract
In this paper we derive some similarity solutions of a nonlinear equation associated with a free boundary problem arising in the shallow-water approximation in glaciology. In addition we present a classical potential symmetry analysis of this second-order nonlinear degenerate parabolic equation related to non- Newtonian ice sheet dynamics in the isothermal case. After obtaining a general result connecting the thickness function of the ice sheet and the solution of the nonlinear equation (without any unilateral formulation), a particular example of a similarity solution to a problem formulated with Cauchy boundary conditions is described. This allows us to obtain several qualitative properties on the free moving boundary in the presence of an accumulation-ablation function with realistic physical properties.
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References
Antontsev, S.N., Diaz, J.I., and Shmarev, S.I., Energy Methods for Free Boundary Problems (Birkhäuser, Boston, 2002).
Bueler, E, Lingle, C.S., Kallen-Brown, J.A., Covey, D.N., AND Bowman L.N. (2005), Exact solutions and verification of numerical models for isothermal ice sheets, J. Glaciol. 51, 291–306.
Bluman, G.W. and Kumei, S., Symmetries and Differential Equations (Springer-Verlag, New York, 1989).
Bluman, G.W., Reid, G.J., and Kumei, S. (1988), New classes of symmetries for partial differential equations, J. Math. Phys. 29, 307–318.
Calvo, N., Diaz, J. I. Durany, J., Schiavi, E., and Vazquez, C. (2002), On a doubly non-linear obstacle problem modelling ice sheet dynamics, SIAM J. Appl. Math. 29, 683–707.
Carrillo, J., and Wittbol, P. (1999), Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Diff. Eq. 156, 93–121.
Diaz, J.I. and Schiavi, S., Tratamiento matematico de una ecuacion parabolica cuasilineal degenerada en Glaciologia, (Electronic Proceedings of the XIV CEDYA-IV Congreso de Matemática Aplicada, Vic, Barcelona, 1995, http://www.mal.upc.es/cedya/cedya.html)
Diaz, J.I. and Schiavi, S. (1999), On a degenerate parabolic/hyperbolic system in glaciology giving rise to free boundary, Nonlinear Anal, 38, 649–673.
Diaz, J.I. and Shmarev, S.I., On the free boundary associate to some nonlinear parabolic equations with discontinuous perturbations. In press: Archive Rational Mechanics and Analysis, 2008.
Duvaut, G. and Lions, J.L., Les Inéquations en Mécanique el en Physique (París, Dunod, 1972).
Fowler, A.C. (1992), Modelling ice sheet dynamics, Geophys. Astrophys. Fluid Dyn. 63, 29–65.
Halifar, P., (1981), On the dynamics of ice sheets, J. Geophys. Res. 86C11, 11065–11072.
Halifar, P., (1983), On the dynamics of ice sheets 2, J. Geophys. Res. 88C10, 6043–6051.
Hindmarsh, R.C.A. (1990), Time-scales and degrees of freedom operating in the evolution of continental ice sheets, Trans R. Soc. Edinburgh, Earth Sci. 81, 371–384.
Hindmarsh, R.C.A., Qualitative Dynamics of Marine Sheets, In Ice in the climate system, Berlin (ed. Peltier, W.R.) (Springer-Verlag, NATO ASI Series I: Global Environmental Change 12, 1993) pp. 67–93.
Nye, J.F., (2000), A flow model for the polar caps of Mars, J. Glaciol. 46, 438–444.
Paterson, W.S.B., The Physics of Glaciers (Pergamon, Oxford, 1981).
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© 2008 Birkhäuser Verlag, Basel
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Díaz, J.I., Wiltshire, R.J. (2008). Potential Symmetry Properties of a Family of Equations Occuring in Ice Sheet Dynamics. In: Camacho, A.G., Díaz, J.I., Fernändez, J. (eds) Earth Sciences and Mathematics. Pageoph Topical Volumes. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9964-1_10
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DOI: https://doi.org/10.1007/978-3-7643-9964-1_10
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