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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. FriersonA. 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. KenigG. 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. KenigG. 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. MelletS. 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. FriersonA. 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. KenigG. 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. KenigG. 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. MelletS. 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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