doi: 10.3934/dcdss.2021165
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Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing

1. 

Depto. Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

2. 

Instituto Matemático Interdisplinar

*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

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

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.

Citation: Gregorio Díaz, Jesús Ildefonso Díaz. Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021165
References:
[1]

D. ArcoyaJ. 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. BeynB. GessP. 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. CaraballoJ. 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. CaraballoJ. 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. DíazJ. 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. DíazG. 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. DíazJ. 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.

show all references

References:
[1]

D. ArcoyaJ. 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. BeynB. GessP. 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. CaraballoJ. 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. CaraballoJ. 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. DíazJ. 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. DíazG. 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. DíazJ. 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.

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