Díaz Díaz, Jesús Ildefonso and Langa, José A. and Valero , José
(2009)
*On the asymptotic behaviour of solutions of a stochastic energy balance climate model.*
Physica D-Nonlinear Phenomena, 238
(9-10).
pp. 880-887.
ISSN 0167-2789

PDF
Restricted to Repository staff only until 31 December 2020. 1MB |

Official URL: http://www.sciencedirect.com/science/article/pii/S0167278909000657

## Abstract

We prove the existence of a random global attractor for the multivalued random dynamical system associated to a nonlinear multivalued parabolic equation with a stochastic term of amplitude of the order of F. The equation was initially suggested by North and Cahalan (following a previous deterministic model proposed by M.I. Budyko), for the modeling of some non-deterministic variability (as, for instance, the cyclones which can be treated as a fast varying component and are represented as a white-noise process) in the context of energy balance climate models. We also prove the convergence (in some sense) of the global attractors, when epsilon -> 0, i.e., the convergence to the global attractor for the associated deterministic case (epsilon = 0).

Item Type: | Article |
---|---|

Uncontrolled Keywords: | partial-differential equations; random dynamical-systems; stationary solutions; attractors; inclusions; climatology model; random global attractor; pullback attractor; stochastic partial differential equation |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 15115 |

References: | G.R. North, R.F. Cahalan, Predictability in a solvable stochastic climate model, J. Atmospheric Sci. 38 (1982) 504–513. M.I. Budyko, The effects of solar radiation variations on the climate of the earth, Tellus 21 (1969) 611–619. P. Imkeller, Energy balance models-viewed from stochastic dynamics, in: P. Imkeller, J.-S. von Storch (Eds.), Stochastic Climate Models, Birkhäuser, Basel, 2001, pp. 213–240. P. Imkeller, J.-S. von Storch, Stochastic Climate Models, Birkhäuser, Basel, 2001. J.I. Díaz, Mathematical analysis of some diffusive energy balance climate models, in: J.I. Díaz, J.L. Lions (Eds.), Mathematics, Climate and Environment, Masson, Paris, 1993, pp. 28–56. J.I. Díaz, L. Tello, A nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collect. Math. 50 (1999) 19–51. R. Bermejo, J. Carpio, J.I. Díaz, L. Tello, Mathematical and numerical analysis of a nonlinear diffusive climate energy balance model, Math. Comput. Modelling (2008). G. Díaz, J.I. Díaz, On a stochastic parabolic PDE arising in climatology, Rev. R. Acad. Cien. Serie A Matem 96 (1) (2002) 123–128. J.I. Díaz, Diffusive energy balance models in climatology, in: D. Cioranescu, J.L. Lions (Eds.), Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. XIV, North-Holland, Amsterdam, 2002, pp. 297–328. J.I. Díaz, L. Tello, Infinitely many stationary solutions for a simple climate model via a shooting method, Math. Methods Appl. Sci. 25 (2002) 327–334. I. Gyöngy, É. Pardoux, On the regularization effect of space-time white noise on quasi-linear parabolic partial differential equations, Probab. Theory Related Fields 97 (1–2) (1993) 211–229. A. Bensoussan, R. Temam, Équations aux dérivées partielles stochastiques non linéaires. Israel J. Math. 11 (1972) 95–129. J.I. Díaz, J. Hernández, L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology, J. Math. Anal. Appl. 216 (1997) 593–613. D. Arcoya, J.I. Díaz, L. Tello, S-shaped bifurcation branch in a quasilinear multivalued model arising in climatology, J. Differential Equations 150 (1998) 215–225. G. Hetzer, The number of stationary solutions for a one-dimensional Budykotype climate model. Nonlinear Anal. RWA 2 (2001) 259–272. R. Bermejo, J. Carpio, J.I. Díaz, P. Galán de Sastre, A finite element algorithm of a nonlinear diffusive climate energy balance model, Pure Appl. Geophys. 165 (6) (2008) 1025–1048. K.Y. Kim, G.R. North, EOF analysis of surface temperature field in a stochastic climate model, J. Clim. 6 (1993) 1681–1690. S. Hu., N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I, Kluwer Academic Publishers, Dordrecht, 1997. C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, in: LNM, vol. 580, Springer, 1977. J.P. Aubin, A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. H. Crauel, F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields 100 (1994) 365–393. T. Caraballo, J.A. Langa, J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal. 48 (2002) 805–829. T. Caraballo, J.A. Langa, J. Valero, Addendum to global attractors for multivalued random dynamical systems, Nonlinear Anal. 61 (2005) 277–279. A.V. Kapustyan, A random attractor of a stochastically perturbed evolution inclusion, Differential Equations 40 (2004) 1383–1388. V.S. Melnik, J. Valero, On Attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal. 6 (1998) 83–111. T. Caraballo, J.A. Langa, J. Valero, Approximation of attractors for multivalued random dynamical systems, Int. J. Math. Game Theory Algebra 11 (4) (2001) 67–92. A.V. Kapustyan, J. Valero, Attractors of multivalued semiflows generated by differential inclusions and their approximations, Abstr. Appl. Anal. 5 (2000) 33–46. A.V. Kapustyan, V.S. Melnik, J. Valero, V.V. Yasinsky, Global Attractors of Multi-Valued Evolution Equations Without Uniqueness, Naukova Dumka, Kiev, 2008. J.P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. H. Brezis, Monotonicity methods in Hilbert spaces and some application to nonlinear partial differential equations, in: E. Zarantonello (Ed.), Contributions to nonlinear functional analysis, Academic Press, New York, 1971. pp. 101–156. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976. L. Arnold, Random dynamical systems, in: Springer Monographs in Mathematics, 1998. I.I. Vrabie, Compactness Methods for Nonlinear Equations, Pitman Longman, London, 1987. A.A. Tolstonogov, Y.I. Umansky, On solutions of evolution inclusions. 2, Sibirsk. Mat. Zh. 33 (4) (1992) 163–173. J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Gauthier-Villar, Paris, 1969. R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1979. T. Caraballo, J.A. Langa, J. Valero, On the relationship between solutions of stochastic and random differential inclusions, Stoch. Anal. Appl. 21 (2003) 545–557. |

Deposited On: | 07 May 2012 08:51 |

Last Modified: | 06 Feb 2014 10:16 |

Repository Staff Only: item control page