Determination of the insolation function in the nonlinear Sellers climate model
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 5, pp. 683-713.

We are interested in the climate model introduced by Sellers in 1969 which takes the form of some nonlinear parabolic equation with a degenerate diffusion coefficient. We investigate here some inverse problem issue that consists in recovering the so-called insolation function. We not only solve the uniqueness question but also provide some strong stability result, more precisely unconditional Lipschitz stability in the spirit of the well-known result by Imanuvilov and Yamamoto (1998) [22]. The main novelties rely in the fact that the considered model is degenerate and above all nonlinear. Indeed we provide here one of the first result of Lipschitz stability in a nonlinear case.

DOI: 10.1016/j.anihpc.2012.03.003
Keywords: Nonlinear parabolic equation, Degenerate diffusion, Climate models, Inverse problems, Carleman estimates, Hardy inequalities
@article{AIHPC_2012__29_5_683_0,
     author = {Tort, J. and Vancostenoble, J.},
     title = {Determination of the insolation function in the nonlinear {Sellers} climate model},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {683--713},
     publisher = {Elsevier},
     volume = {29},
     number = {5},
     year = {2012},
     doi = {10.1016/j.anihpc.2012.03.003},
     zbl = {1270.35283},
     mrnumber = {2971027},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.03.003/}
}
TY  - JOUR
AU  - Tort, J.
AU  - Vancostenoble, J.
TI  - Determination of the insolation function in the nonlinear Sellers climate model
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2012
DA  - 2012///
SP  - 683
EP  - 713
VL  - 29
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.03.003/
UR  - https://zbmath.org/?q=an%3A1270.35283
UR  - https://www.ams.org/mathscinet-getitem?mr=2971027
UR  - https://doi.org/10.1016/j.anihpc.2012.03.003
DO  - 10.1016/j.anihpc.2012.03.003
LA  - en
ID  - AIHPC_2012__29_5_683_0
ER  - 
%0 Journal Article
%A Tort, J.
%A Vancostenoble, J.
%T Determination of the insolation function in the nonlinear Sellers climate model
%J Annales de l'I.H.P. Analyse non linéaire
%D 2012
%P 683-713
%V 29
%N 5
%I Elsevier
%U https://doi.org/10.1016/j.anihpc.2012.03.003
%R 10.1016/j.anihpc.2012.03.003
%G en
%F AIHPC_2012__29_5_683_0
Tort, J.; Vancostenoble, J. Determination of the insolation function in the nonlinear Sellers climate model. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 5, pp. 683-713. doi : 10.1016/j.anihpc.2012.03.003. http://www.numdam.org/articles/10.1016/j.anihpc.2012.03.003/

[1] L. Baudouin, J.P. Puel, An inverse problem for the Schrödinger equation, Inverse Problems 18 no. 6 (2002), 1537-1554 | MR | Zbl

[2] A. Benabdallah, P. Gaitan, J. Le Rousseau, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim. 46 no. 5 (2007), 1849-1881 | MR | Zbl

[3] A. Benabdallah, Y. Dermenjian, J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math. Anal. Appl. 336 (2007), 865-887 | MR | Zbl

[4] A. Bensoussan, G. Da Prato, M.C. Delfour, S.K. Mitter, Representation and Control of Infinite-Dimensional Systems, vol. 1, Systems Control Found. Appl., Birkhäuser Boston, Inc., Boston, MA (1992) | MR

[5] R. Bermejo, J. Carpio, J.I. Diaz, L. Tello, Mathematical and numerical analysis of a non linear diffusive climate energy balance model, Math. Comput. Modelling 49 (2009), 1180-1210 | MR | Zbl

[6] A.L. Bukhgeim, M.V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR 260 no. 2 (1981), 269-272 | MR

[7] M. Campiti, G. Metafune, D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum 57 no. 1 (1998), 1-36 | MR | Zbl

[8] P. Cannarsa, P. Martinez, J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim. 47 no. 1 (2008), 1-19 | MR | Zbl

[9] P. Cannarsa, P. Martinez, J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations 10 no. 2 (2005), 153-190 | MR | Zbl

[10] P. Cannarsa, P. Martinez, J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations, Commun. Pure Appl. Anal. 3 no. 4 (2004), 607-635 | MR | Zbl

[11] P. Cannarsa, D. Rocchetti, J. Vancostenoble, Generation of analytic semi-groups in L 2 for a class of degenerate elliptic operators, Control Cybernet. 37 no. 4 (2008), 831-878 | EuDML | MR | Zbl

[12] P. Cannarsa, P. Martinez, J. Vancostenoble, Carleman estimates for degenerate parabolic operators with applications, Mem. Amer. Math. Soc., in press. | MR

[13] P. Cannarsa, J. Tort, M. Yamamoto, Determination of a source term in a degenerate parabolic equation, Inverse Problems 26 no. 10 (2010), 105003 | MR | Zbl

[14] M. Cristofol, L. Roques, Biological invasions: deriving the regions at risk from partial measurements, Math. Biosci. 215 no. 2 (2008), 158-166 | MR | Zbl

[15] R. Dautray, J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Tome 3, Masson, Paris (1985) | MR | Zbl

[16] J.I. Diaz, Mathematical analysis of some diffusive energy balance models in climatology, J.I. Diaz, J.L. Lions (ed.), Mathematics, Climate and Environnement, Masson (1993), 28-56 | MR | Zbl

[17] J.I. Diaz, On the Mathematical Treatment of Energy Balance Climate Models, NATO ASI Ser. Ser. I Glob. Environ. Change vol. 48, Springer, Berlin (1997) | MR | Zbl

[18] J.I. Diaz, Diffusive Energy Balance Models in Climatology, Stud. Math. Appl. vol. 31, North-Holland, Amsterdam (2002) | MR | Zbl

[19] H. Egger, H.W. Engl, M.V. Klibanov, Global uniqueness and Hölder stability for recovering a non linear source term in a parabolic equation, Inverse Problems 21 (2005), 271-290 | MR | Zbl

[20] A.V. Fursikov, O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser. vol. 34, Seoul National University, Seoul, Korea (1996) | MR | Zbl

[21] G. Hetzer, The number of stationary solutions for a one-dimensional Budyko-type climate model, Nonlinear Anal. Real World Appl. 2 (2001), 259-272 | MR | Zbl

[22] O.Y. Imanuvilov, M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimates, Inverse Problems 14 no. 5 (1998), 1229-1245 | MR | Zbl

[23] V. Isakov, Inverse Problems for Partial Differential Equations, Appl. Math. Sci. vol. 127, Springer-Verlag, New York (1998) | MR | Zbl

[24] M.V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems 8 no. 4 (1992), 575-596 | MR | Zbl

[25] M.V. Klibanov, Global uniqueness of a multidimensional inverse problem for a nonlinear parabolic equation by a Carleman estimate, Inverse Problems 20 (2004), 1003-1032 | MR | Zbl

[26] M.V. Klibanov, A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse Ill-posed Probl. Ser., VSP, Utrecht (2004) | MR | Zbl

[27] J.L. Lions, Equations différentielles opérationnelles et problèmes aux limites, Grundlehren Math. Wiss. vol. 111, Springer-Verlag, Berlin–Göttingen–Heidelberg (1961) | MR | Zbl

[28] J.L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1, Travaux et Rech. Math. vol. 17, Dunod, Paris (1968) | MR | Zbl

[29] A. Lunardi, Analytic Semigroups and Optimal Regularity Results, Progr. Nonlinear Differential Equations Appl. vol. 16, Birkhäuser, Basel (1995)

[30] G.R. North, J.G. Mengel, D.A. Short, Simple energy balance model resolving the season and continents: applications to astronomical theory of ice ages, J. Geophys. Res. 88 (1983), 6576-6586

[31] P. Martinez, J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ. 6 no. 2 (2006), 325-362 | MR | Zbl

[32] J.P. Puel, M. Yamamoto, Applications of exact controllability to some inverse problems for the wave equation, Control of Partial Differential Equations and Applications, Laredo, 1994, Lect. Notes Pure Appl. Math. vol. 174, Dekker, New York (1996), 241-249 | Zbl

[33] J. Tort, Determination of source terms in a degenerate parabolic equation from a locally distributed observation, C. R. Acad. Sci. Paris Sér. I 348 no. 23–24 (2010), 1287-1291 | MR | Zbl

[34] J. Vancostenoble, Improved Hardy–Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst. Ser. S 4 no. 3 (2011), 761-790 | MR | Zbl

[35] J. Vancostenoble, Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems, C. R. Acad. Sci. Paris Sér. I 348 (2010), 801-805 | MR | Zbl

[36] J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Comm. Partial Differential Equations 36 no. 8 (2011), 1287-1317 | MR | Zbl

[37] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems 25 no. 12 (2009), 123013 | MR | Zbl

[38] M. Yamamoto, J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems 17 no. 4 (2001), 1181-1202 | MR | Zbl

Cited by Sources: