Complutense University Library

Lagrangian approach to the study of level sets II: A quasilinear equation in climatology

Díaz Díaz, Jesús Ildefonso and Shmarev, Sergey (2009) Lagrangian approach to the study of level sets II: A quasilinear equation in climatology. Journal of Mathematical Analysis and Applications, 352 (1). pp. 475-495. ISSN 0022-247X

[img] PDF
Restricted to Repository staff only until 31 December 2020.

245kB

Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X08009426

View download statistics for this eprint

==>>> Export to other formats

Abstract

We study the dynamics and regularity of the level sets in solutions of the semilinear parabolic equation u(t) - Delta p(u) + f is an element of aH(u - mu) in Q = Omega x (0, T], P is an element of (1, infinity), where Omega subset of R(n) is a ring-shaped domain, Delta(p)u is the p-Laplace operator, a and mu are given positive constants, and H(.) is the Heaviside maximal monotone graph: H(s) = 1 if s > 0, H(0) = [0, 1], H(s) = 0 if s < 0. The mathematical models of this type arise in climatology, the case p = 3 was proposed and justified by P. Stone in 1972. We establish the conditions on the initial data which guarantee that the level sets Gamma(mu)(t) = {x: u(x, t) = mu} are hypersurfaces, study the regularity of Gamma(mu)(t) and derive the differential equation that governs the dynamics of Gamma(mu)(t). The analysis is based on the introduction of a system of Lagrangian coordinates that transforms the moving surface Gamma(mu)(t) into a stationary one.


Item Type:Article
Uncontrolled Keywords:regularity; parabolic p-laplacian; lagrangian coordinates; climatic energy balance models; free boundary problem
Subjects:Sciences > Mathematics > Differential equations
ID Code:15117
References:

M. Budyko, The effects of solar radiation variations on the climate of the earth, Tellus 21 (1969) 611–619.

P. Daskalopoulos, R. Hamilton, C∞-regularity of the interface of the evolution p-Laplacian equation, Math. Res. Lett. 5 (1998) 685–701.

P. Daskalopoulos, R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc. 11 (1998) 899–965.

J. Díaz, Mathematical analysis of some diffusive energy balance climate models, in: J. Díaz, J. Lions (Eds.), Mathematics, Climate and Environment, Masson, Paris, 1993, pp. 28–56.

J. Díaz, L.L. Tello, A nonlinear parabolic problem on a Riemannian manifold without boundary arising in climatology, Collect. Math. 50 (1999) 19–51.

J. Díaz, S. Shmarev, Lagrangian approach to the study of level sets: Application to a free boundary problem in climatology, Arch. Ration. Mech. Anal., doi: 10.1007/s00205–008-0164-y, in press.

R. Gianni, J. Hulshof, The semilinear heat equation with a Heaviside source term, European J. Appl. Math. 3 (1992) 367–379.

R. Gianni, R. Ricci, Classical solvability of some free boundary problems through the geometry of the level lines, Adv. Math. Sci. Appl. 5 (1995) 557–567.

L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982, translated from the Russian by Howard L. Silcock.

O.A. Ladyženskaja, V.A. Solonnikov, N.N. Uralʾceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., vol. 23, Amer. Math. Soc., Providence, RI, 1967, translated from the Russian by S. Smith.

O.A. Ladyzhenskaya, N.N. Uralʾtseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968, translated from the Russian by Scripta Technica, Inc., translation editor: Leon Ehrenpreis.

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, 1969.

J. Simon, Compact sets in the space Lp(0,T;B), Ann. Mat. Pura Appl. (4) 146 (1987) 65–96.

P. Stone, A simplified radiative-dynamical model for the static stability of rotating atmospheres, J. Atmospheric Sci. 29 (1972) 405–418.

X. Xu, Existence and regularity theorems for a free boundary problem governing a simple climate model, Appl. Anal. 42 (1991) 33–57.

Deposited On:07 May 2012 08:35
Last Modified:06 Feb 2014 10:16

Repository Staff Only: item control page