Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing

Gregorio Díaz; Jesús Ildefonso Díaz

doi:10.3934/dcdss.2021165

Article Contents

2022, Volume 15, Issue 10: 2837-2870. Doi: 10.3934/dcdss.2021165

This issue Previous Article A backward SDE method for uncertainty quantification in deep learning Next Article Bounded positive solutions for diffusive logistic equations with unbounded distributed limitations

Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing

Gregorio Díaz^1, and
Jesús Ildefonso Díaz^1,2, ,

1.
Depto. Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2.
Instituto Matemático Interdisplinar, Depto. Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

^*Corresponding author: Jesús Ildefonso Díaz
^*Corresponding author: Jesús Ildefonso Díaz

Dedicated to Georg Hetzer on occasion of his 75th birthday

Received: October 2021

Revised: November 2021

Early access: December 2021

Published: October 2022

Partially supported the UCM Research Group MOMAT (ref. 910480) and the projects MTM2017-85449-P and PID2020-112517GB-I00 of the DGISPI, Spain.

Abstract / Introduction Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

We consider a class of one-dimensional nonlinear stochastic parabolic problems associated to Sellers and Budyko diffusive energy balance climate models with a Legendre weighted diffusion and an additive cylindrical Wiener processes forcing. Our results use in an important way that, under suitable assumptions on the Wiener processes, a suitable change of variables leads the problem to a pathwise random PDE, hence an essentially "deterministic" formulation depending on a random parameter. Two applications are also given: the stability of solutions when the Wiener process converges to zero and the asymptotic behaviour of solutions for large time.

Keywords:

Mathematics Subject Classification: 60H15, 60H30, 35R60, 86A08.

Citation:

Full Text(HTML)

Figure 1. Values of the parameter $ {{\rm{Q}}} $ with different multiplicity

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	D. Arcoya, J. I. Díaz and L. Tello, S-Shaped bifurcation branch in a quasilinear multivalued model arising in climatology, J. Differential Equations, 150 (1998), 215-225. doi: 10.1006/jdeq.1998.3502.
[2]	L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[3]	V. Barbu, Nonlinear Differential Equations of MonotoneType in Banach Spaces, SpringerMonographs in Mathematics. Springer, 2010. doi: 10.1007/978-1-4419-5542-5.
[4]	V. Barbu and M. Röckner, An operational approach to stochastic differential equations driven by linear multiplicative noise, J. Eur. Math. Soc., 17 (2015), 1789-1815. doi: 10.4171/JEMS/545.
[5]	P. Bénilan, M. G. Crandall and A. Pazy, Nonlinear Evolution Equations Governed by Accretive Operators, Manuscript of Book in Preparation.
[6]	S. Bensid and J. I. Díaz, On the exact number of monotone solutions of a simplified Budyko climate model and their different stability, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1033-1047. doi: 10.3934/dcdsb.2019005.
[7]	A. Bensoussan and R. Temam, ćquations aux derivées partielles stochastiques non lineaires, Israel J. Math., 11 (1972), 95-129. doi: 10.1007/BF02761449.
[8]	W. J. Beyn, B. Gess, P. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469. doi: 10.1080/03605302.2010.523919.
[9]	H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.
[10]	Z. Brzezniak and J. van Neerven, Stochastic convolution is separable Bancah spaces and the stochastic lineal Cauchy problem, Studia Math., 143 (2000), 43-74. doi: 10.4064/sm-143-1-43-74.
[11]	R. Buckdahn and É. Pardoux, Monotonicity methods for white noise driven quasilinear SPDEs, Diffusion Processes and Related Problems in Analysis, I, M. Pinsky, ed., Birkhäuser Boston, MA, 22 (1990), 219–233.
[12]	M. I. Budyko, The effect of solar radiation variations on the climate of the Earth, Tellus, 21 (1969), 611-619.
[13]	T. Caraballo, J. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805-829. doi: 10.1016/S0362-546X(00)00216-9.
[14]	T. Caraballo, J. A. Langa and J. Valero, On the relationship between solutions of stochastic and random differential inclusions, Stoch. Anal. Appl., 21 (2003), 545-557. doi: 10.1081/SAP-120020425.
[15]	A. N. Carvalho, J. A. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.
[16]	P.-L. Chow, Stochastic Partial Differential Equation, CRC Press, Boca Raton, 2015.
[17]	H. Crauel and F. Flandolfi, Attractors for random dynamical systems, Prob. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[18]	G. Da Prato, Kolmogorov Equations for Stochastic PDEs, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7909-5.
[19]	G. Da Prato and H. Frankowska, A stochastic Filippov theorem, Stochastic Anal. Appl., 12 (1994), 409-426. doi: 10.1080/07362999408809361.
[20]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Presss, 1992. doi: 10.1017/CBO9780511666223.
[21]	G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Presss, 1996. doi: 10.1017/CBO9780511662829.
[22]	G. Díaz and J. I. Díaz, On a stochastic parabolic PDE arising in Climatology, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 96 (2002), 123-128.
[23]	J. I. Díaz, Mathematical analysis of some diffusive energy balance climate models, In Mathematics, Climate and Environment (J.I. Díaz and J.-L. Lions, eds.), Masson, Paris, 28–56, 1993.
[24]	J. I. Díaz, J. Hernández and 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. doi: 10.1006/jmaa.1997.5691.
[25]	J. I. Díaz and G. Hetzer, A functional quasilinear reaction-diffusion equation arising in climatology, In ćquations Aux Dérivées Partielles et Applications, Articles dédiés à J.-L. Lions, Gauthier-Villars, Elsevier, Paris (1998), 461–480.
[26]	J. I. Díaz, G. Hetzer and L. Tello, An energy balance climate model with hysteresis, Nonlinear Anal., 64 (2006), 2053-2074. doi: 10.1016/j.na.2005.07.038.
[27]	J. I. Díaz, J. A. Langa and J. Valero, On the asymptotic behaviour of solutions of a stochastic energy balance climate model, Phys. D, 238 (2009), 880-887. doi: 10.1016/j.physd.2009.02.010.
[28]	J. I. Díaz and L. Tello, On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in climatology, Collect. Math., 50 (1999), 19-51.
[29]	J. I. Díaz and I. I. Vrabie, Existence for reaction-diffusion systems. A compactness method approach, J. Math. Anal. Appl., 188 (1994), 521-540. doi: 10.1006/jmaa.1994.1443.
[30]	H. Doss, Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré Sect. B (N. S.), 13 (1977), 99-125.
[31]	I. Gyöngy and E. Pardoux, On the regularization effecto to space-time white noise on quasi-linear stochastic partial differential equations, Probab. Theory Relat. Fields, 97 (1993), 211-229. doi: 10.1007/BF01199321.
[32]	X. Han and P. E. Kloeden, Stochastic Ordinary Differential Equations and Their Numerical Solutions, Springer Singapore, 2017. doi: 10.1007/978-981-10-6265-0.
[33]	X. Han and P. E. Kloeden, Sigmoidal approximations of Heaviside functions in neural lattice models, J. Differential Equations, 268 (2020), 5283-5300. doi: 10.1016/j.jde.2019.11.010.
[34]	X. Han and P. E. Kloeden, Corrigendum to "Sigmoidal approximations of Heaviside functions in neural lattice models", J. Differential Equations, 274 (2020), 1214-1220. doi: 10.1016/j.jde.2020.11.017.
[35]	G. Hetzer, The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston J. Math., 16 (1990), 203-216.
[36]	G. Hetzer, S-shapedness for energy balance climate models of Sellers-type, In The Mathematics of Models for Climatology and Environment (J. I. Díaz, ed.), Springer, Berlin, (1997), 253–287.
[37]	G. Hetzer, The number of stationary solntions for one-dimensional Budyko-type climate models, Nonlinear Anal. Real World Appl., 2 (2001), 259-272. doi: 10.1016/S0362-546X(00)00103-6.
[38]	G. Hetzer and P. Schmidt, Analysis of energy balance models, World Congress of Nonlinear Analysts '92, (1996), 1609–1618.
[39]	P. Imkeller, Energy balance models-viewed from stochastic dynamics, In Stochastic Climate Models (P. Imkeller and J.-S. von Storch, eds.), Birkhäuser, Basel, (2001), 213–240.
[40]	H. G. Kaper and H. Engler, Mathematics and Climate, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania, 2013. doi: 10.1137/1.9781611972610.
[41]	A. V. Kapustyan, A random attractor of a stochastically perturbed evolution inclusion, Differ. Equ., 40 (2004), 1383-1388. doi: 10.1007/s10625-005-0060-2.
[42]	K. Liu, Stochastic Stability of Differential Equations in Abstract Spaces, Cambridge University Pres, 2019. doi: 10.1017/9781108653039.
[43]	W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext. Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.
[44]	V. Lucarini, L. Serdukova and G. Margazoglou, Lévy-noise versus Gaussian-noise-induced Transitions in the Ghil-Sellers Energy Balance Model, Nonlinear Processes in Geophysics, 2021. doi: 10.5194/npg-2021-34.
[45]	A. F. Nikiforov, S. K. Suslov and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer-Verlag, New York, 1991 doi: 10.1007/978-3-642-74748-9.
[46]	G. R. North and R. F. Cahalan, Predictability in a solvable stochastic climate model, J. Atmospheric Science, 38 (1981), 504-513. doi: 10.1175/1520-0469(1981)038<0504:PIASSC>2.0.CO;2.
[47]	G. R. North and K. Y. Kim, Energy Balance Climate Models, Wiley-VCH, Weinheim, Germany, 2017.
[48]	B. Schmidt, Bifurcation of Stationary Solutions for Legendre-Type Boundary Value Problems Arising From Climate Modeling, Thesis (Ph.D.)–Auburn University, 1994.
[49]	W. S. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, J. Appl. Meteorol, 8 (1969), 392-400. doi: 10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.
[50]	H. J. Sussmann, On the gap between deterministic and stochastic differential equations, Ann. Probability, 6 (1978), 19-41.
[51]	J. van Neerven and M. Veraar, Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes, Stoch PDE: Anal Comp, 2021. doi: 10.1007/s40072-021-00204-y.
[52]	I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, 2$^{nd}$ edition, Pitman Monographs and Surveys in Pure and Applied Mathematics, New York, 1995.
[53]	S. Yotsutani, Evolution equations associated with the subdifferentials, J. Math. Soc. Japan, 31 (1979), 623-646. doi: 10.2969/jmsj/03140623.