Riv. Mat. Univ. Parma, Vol. 7, No. 2, 2016

Patrick Martinez[a], Jacques Tort[b] and Judith Vancostenoble[c]

Lipschitz stability for an inverse problem for the 2D-Sellers model on a manifold

Pages: 351-389
Received: 11 February 2016
Accepted: 24 January 2017
Mathematics Subject Classification (2010): 58J35, 35K55.
Keywords: PDEs on manifolds, nonlinear parabolic equations, climate models, inverse problems, Carleman estimates.
Author address:
[a], [b], [c]: University of Toulouse 3, 118 route de Narbonne 31062 Toulouse Cedex 9, France

Abstract: In this paper, we are interested in some inverse problem that consists in recovering the so-called insolation function in the 2-D Sellers model on a Riemannian manifold that materializes the Earth's surface. For this nonlinear problem, we obtain a Lipschitz stability result in the spirit of the result by Imanuvilov-Yamamoto in the case of the determination of the source term in the linear heat equation. The paper complements an analogous study by Tort-Vancostenoble in the case of the 1-D Sellers model.


[1] W. Abikoff, The uniformization theorem, Amer. Math. Monthly 88 (1981), no. 8, 574-592. MR0628026
[2] T. Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Springer-Verlag, New-York 1982. MR0681859
[3] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and control of infinite-dimensional systems, Vol.1, Systems & Control: Foundations & Applications, Birkhäuser Boston 1992. MR1182557
[4] R. Bermejo, J. Carpio, J. I. Diaz and L. Tello, Mathematical and numerical analysis of a nonlinear diffusive climate energy balance model, Math. Comput. Modelling 49 (2009), 1180-1210. MR2495033
[5] H. Brezis, Analyse Fonctionnelle, Dunod, Paris 1999.
[6] M.M. Cavalcanti, V.N. Domingos Cavalcanti, R. Fukuoka and J.A.Soriano, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result, Arch. Ration. Mech. Anal. 197 (2010), no. 3, 925-964. MR2679361
[7] P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications, Mem. Amer. Math. Soc. 239 (2016), no. 1133, 209 pp. MR3430764
[8] T. Cazenave and A. Haraux, An introduction to semilinear evolution equations, Oxford Lecture Ser. Math. Appl. 13, Oxford University Press, New York 1998. MR1691574
[9] I. Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, 115, Academic Press Inc., Orlando, FL 1984. MR0768584
[10] I. Chavel, Isoperimetric inequalities, Differential geometric and analytic perspectives, Cambridge Tracts in Mathematics, 145, Cambridge University Press, Cambridge 2001. MR1849187
[11] M. Cristofol and L. Roques, Stable estimation of two coefficients in a nonlinear Fisher-KPP equation, Inverse Problems 29 (2013), no. 9, 095007, 18 pp. MR3094488
[12] R. Dautray and J.-L. Lions, Analyse mathématiques et calcul numérique pour les sciences et les techniques, Tome 3, Masson, Paris 1985. MR0902802
[13] J. I. Diaz, Mathematical analysis of some diffusive energy balance models in climatology, Mathematics, climate and environment (Madrid, 1991), J. I. Diaz and J.-L. Lions, eds., RMA Res. Notes Appl. Math., 27, Masson, Paris 1993, 28-56. MR1263042
[14] J. I. Diaz, On the mathematical treatment of energy balance climate models, NATO ASI Ser. Ser. I Glob. Environ. Change, 48, Springer, Berlin 1997, 217-251. MR1635284
[15] J. I. Diaz, Diffusive energy balance models in climatology, Stud. Math. Appl., 31, North-Holland, Amsterdam 2002, 297-328. MR1935999
[16] J. I. Diaz, G. Hetzer and L. Tello, An energy balance climate model with hysteresis, Nonlinear Anal. 64 (2006), 2053-2074. MR2211199
[17] H. Egger, H. W. Engl and M. V. Klibanov, Global uniqueness and Hölder stability for recovering a nonlinear source term in a parabolic equation, Inverse Problems 21 (2005), 271-290. MR2146176
[18] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations, Lecture Notes Ser., 34, Seoul National University, Seoul, Korea 1996. MR1406566
[19] S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry, Third edition, Universitext, Springer-Verlag, Berlin 2004. MR2088027
[20] G. Hetzer, The number of stationary solutions for a one-dimensional Budyko-type climate model, Nonlinear Anal. Real World Appl. 2 (2001), 259-272. MR1822422
[21] O. Yu. Imanuvilov, Controllability of parabolic equations (Russian), translated from Mat. Sb. 186 (1995), no. 6, 109-132, Sb. Math. 186 (1995), no. 6, 879-900. MR1349016
[22] O. Yu. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimates, Inverse Problems 14 (1998), no. 5, 1229-1245. MR1654631
[23] V. Isakov, Inverse Problems for Partial Differential Equations, Second edition, Appl. Math. Sci., 127, Springer, New York 2006. MR2193218
[24] J. Lafontaine, Introduction aux variétés différentielles (French), Presses Universitaires de Grenoble, Grenoble 1996, 299 pp. ZbMath
[25] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur (French), Comm. Partial Differential Equations 20 (1995), 335-356. MR1312710
[26] J.-L. Lions, Equations différentielles opérationnelles et problèmes aux limites (French), Die Grundlehren des mathematischen Wissenschaften, 111, Springer-Verlag, Berlin-Göttingen-Heidelberg 1961. MR0153974
[27] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol.1 (French), Travaux et Rech. Math., No. 17 Dunod, Paris 1968. MR0247243
[28] A. Lunardi, Analytic semigroups and optimal regularity results, Progress in Nonlinear Differential Equations and their Applications 16, Birkhäuser, Basel, 1995.
[29] L. Miller, Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds, Math. Res. Lett. 12 (2005), no. 1, 37-47. MR2122728
[30] G. R. North, J. G. Mengel and D. A. Short, simple energy balance model resolving the season and continents: applications to astronomical theory of ice ages, J. Geophys. Res. Oceans 88 (1983), 6576-6586. DOI: 10.1029/JC088iC11p06576
[31] F. Punzo, Uniqueness for the heat equation in Riemannian manifolds, J. Math. Anal. Appl. 424 (2015), no. 1, 402-422. MR3286568
[32] F. Punzo, Global existence of solutions to the semilinear heat equation on Riemannian manifolds with negative sectional curvature, Riv. Mat. Univ. Parma (N.S.) 5 (2014), no. 1, 113-138. MR3289599
[33] L. Roques and M. Cristofol, On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation, Nonlinearity 23 (2010), no. 3, 675-686. MR2593914
[34] L. Roques, M. D. Checkroun, M. Cristofol, S. Soubeyrand and M.Ghil, Parameter estimation for energy balance models with memory, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), no. 2169, 20140349, 20 pp. MR3238186
[35] M. E. Taylor, Partial Differential Equations I. Basic theory, Second edition, Applied Mathematical Sciences, 115, Springer, New York 2011. MR2744150
[36] M. E. Taylor, Partial Differential Equations II. Qualitative studies in linear equations, Second edition, Applied Mathematical Sciences, 116, Springer-Verlag, New York 2011. MR2743652
[37] M. E. Taylor, Partial Differential Equations III. Nonlinear equations, Second edition, Applied Mathematical Sciences, 117, Springer-Verlag, New York 2011. MR2744149
[38] J. Tort, Problèmes inverses pour des équations paraboliques issues de modèles de climat, Université Toulouse 3 Paul Sabatier, PhD Thesis, 29 Juin 2012. url: http://thesesups.ups-tlse.fr/1649/
[39] J. Tort and J. Vancostenoble, Determination of the insolation function in the nonlinear climates Sellers model, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), 683-713.
[40] S. Weitkamp, A new proof of the uniformization theorem, Ann. Global Anal. Geom. 27 (2005), no. 2, 157-177. MR2131911
[41] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems 25 (2009), no. 12, 123013, 75 pp. MR3460049
[42] M. Yamamoto and J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems 17 (2001), no. 4, 1181-1202. MR1861508

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