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January  2019, 24(1): 211-229. doi: 10.3934/dcdsb.2018102

## Global regularity results for the climate model with fractional dissipation

 1 School of Mathematical Sciences, Anhui University, Hefei 230601, China 2 Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA 3 School of Mathematics and computation Sciences, Anqing Normal University, Anqing 246133, China

* Corresponding author: Jiahong Wu

Received  January 2017 Revised  January 2018 Published  January 2019 Early access  March 2018

Fund Project: B. Dong was partially supported by the NNSFC (No. 11271019, No. 11571240). J. Wu was supported by NSF grant DMS 1614246 and the AT & T Foundation at Oklahoma State University. H. Zhang was as partially supported by the Research Fund of SMS at Anhui University.

This paper studies the global well-posedness problem on a tropical climate model with fractional dissipation. This system allows us to simultaneously examine a family of equations characterized by the fractional dissipative terms $(-Δ)^{\mathit{\alpha }}u$ in the equation of the barotropic mode $u$ and $(-Δ)^β v$ in the equation of the first baroclinic mode $v$. We establish the global existence and regularity of the solutions when the total fractional power is 2, namely ${\mathit{\alpha }}+ β = 2$.

Citation: Boqing Dong, Wenjuan Wang, Jiahong Wu, Hui Zhang. Global regularity results for the climate model with fractional dissipation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 211-229. doi: 10.3934/dcdsb.2018102
##### References:
 [1] S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407. [2] H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, 2011. [3] J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1976. [4] B. Dong, J. Wu, X. Xu and Z. Ye, Global regularity for the 2D micropolar equations with fractional dissipation, submitted for publication. [5] D. Frierson, A. Majda and O. Pauluis, Large scale dynamics of precipitation fronts in the tropical atmosphere: A novel relaxation limit, Comm. Math. Sci., 2 (2004), 591-626.  doi: 10.4310/CMS.2004.v2.n4.a3. [6] M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.  doi: 10.1002/cpa.20253. [7] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. American Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0. [8] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405. [9] J. Li and E. S. Titi, Global well-posedness of the 2D Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 220 (2016), 983-1001.  doi: 10.1007/s00205-015-0946-y. [10] J. Li and E. S. Titi, Global well-posedness of strong solutions to a tropical climate model, Discrete Contin. Dyn. Syst., 36 (2016), 4495-4516.  doi: 10.3934/dcds.2016.36.4495. [11] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Rat. Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2. [12] C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and its Applications in Partial Differential Equations of Fluid Dynamics, Science Press, Beijing, China, 2012 (in Chinese). [13] T. Runst and W. Sickel, Sobolev Spaces of fractional order, Nemytskij operators and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, New York, 1996. [14] Z. Ye, Global regularity for a class of 2D tropical climate model, J. Math. Anal. Appl., 446 (2017), 307-321.  doi: 10.1016/j.jmaa.2016.08.053.

show all references

##### References:
 [1] S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407. [2] H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, 2011. [3] J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1976. [4] B. Dong, J. Wu, X. Xu and Z. Ye, Global regularity for the 2D micropolar equations with fractional dissipation, submitted for publication. [5] D. Frierson, A. Majda and O. Pauluis, Large scale dynamics of precipitation fronts in the tropical atmosphere: A novel relaxation limit, Comm. Math. Sci., 2 (2004), 591-626.  doi: 10.4310/CMS.2004.v2.n4.a3. [6] M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.  doi: 10.1002/cpa.20253. [7] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. American Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0. [8] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405. [9] J. Li and E. S. Titi, Global well-posedness of the 2D Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 220 (2016), 983-1001.  doi: 10.1007/s00205-015-0946-y. [10] J. Li and E. S. Titi, Global well-posedness of strong solutions to a tropical climate model, Discrete Contin. Dyn. Syst., 36 (2016), 4495-4516.  doi: 10.3934/dcds.2016.36.4495. [11] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Rat. Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2. [12] C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and its Applications in Partial Differential Equations of Fluid Dynamics, Science Press, Beijing, China, 2012 (in Chinese). [13] T. Runst and W. Sickel, Sobolev Spaces of fractional order, Nemytskij operators and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, New York, 1996. [14] Z. Ye, Global regularity for a class of 2D tropical climate model, J. Math. Anal. Appl., 446 (2017), 307-321.  doi: 10.1016/j.jmaa.2016.08.053.
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