American Institute of Mathematical Sciences

Advanced
September  2016, 9(3): 551-570. doi: 10.3934/krm.2016006

Local well-posedness for the tropical climate model with fractional velocity diffusion

Caochuan Ma 1, , Zaihong Jiang 1, and  Renhui Wan 2,

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China
2. 

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, Zhejiang, China

Received  September 2015 Revised  December 2015 Published  May 2016

This paper deals with the Cauchy problem for tropical climate model with the fractional velocity diffusion which was derived by Frierson-Majda-Pauluis in [16]. We establish the local well-posedness of strong solutions to this generalized model.
Keywords: local well-posedness., Tropical climate model, fractional velocity diffusion.
Mathematics Subject Classification: Primary: 35Q35, 35Q85; Secondary: 35B65, 76W0.
Citation: Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic and Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006
References:
[1]

P. W. Bates, On some nonlocal evolution equations arising in materials science, in: Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., 48 (2006), 13-52.

[2]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66451-9.

[4]

P. Biler, G. Karch and W. A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 613-637. doi: 10.1016/S0294-1449(01)00080-4.

[5]

J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s \rightarrow 1$ and applications, J. Anal. Math., 87 (2002), 77-101. doi: 10.1007/BF02868470.

[6]

H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ. , 1 (2001), 387-404. doi: 10.1007/PL00001378.

[7]

L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119.

[8]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.

[9]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.

[10]

C. Cao, S. Ibrahim, K. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482. doi: 10.1007/s00220-015-2365-1.

[11]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, 2004. doi: 10.1201/9780203485217.

[12]

W. Craig and P. A. Worfolk, An integrable normal form for water waves in infinite depth, Phys. D, 84 (1995), 513-531. doi: 10.1016/0167-2789(95)00067-E.

[13]

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66165-5.

[14]

C. Fefferman and R. de la Llave, Relativistic stability of matter, I. Rev. Mat. Iberoamericana, 2 (1986), 119-213. doi: 10.4171/RMI/30.

[15]

C. L. Feffermana, D. S. McCormick, J. C. Robinsonb and J. L. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD and related models, J. Funct. Anal., 267 (2014), 1035-1056. doi: 10.1016/j.jfa.2014.03.021.

[16]

D. M. W. Frierson, A. J. Majda and O. M. Pauluis, Large scale dynamics of precipitation fronts in the tropical atmosphere: A novel relaxation limit, Commun. Math. Sci., 2 (2004), 591-626. doi: 10.4310/CMS.2004.v2.n4.a3.

[17]

Z. Jiang and Y. Zhou, Local existence for the generalized MHD equations,, preprint., (). 

[18]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0.

[19]

A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.

[20]

G. M. Kobelkov, Existence of a solution in the large for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. doi: 10.1016/j.crma.2006.04.020.

[21]

I. Kukavica, R. Temam, V. C. Vicol and M. Ziane, Existence and uniqueness of solutions for the hydrostatic Euler equations on a bounded domain with analytic data, C. R. Math. Acad. Sci. Paris, 348 (2010), 639-645. doi: 10.1016/j.crma.2010.03.023.

[22]

I. Kukavica, R. Temam, V. C. Vicol and M. Ziane, Local existence and uniqueness for the hydrostatic euler equations on a bounded domain, J. Differential Equations, 250 (2011), 1719-1746. doi: 10.1016/j.jde.2010.07.032.

[23]

J. Li and E. S. Titi, Global well-posedness of strong solutions to a tropical climate model,, preprint, (). 

[24]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equationsof the atmosphere and appliations, Nonlinearity, 5 (1992), 237-288.

[25]

J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002.

[26]

J. L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAO III), J. Math. Pures Appl., 74 (1995), 105-163.

[27]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511613203.

[28]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 77pp. doi: 10.1016/S0370-1573(00)00070-3.

[29]

C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and Its Applications in Partial Differential Equations of Fluid Dynamics (in Chinese), Science Press, Beijing, China, 2012.

[30]

D. P. Nicholls and M. Taber, Joint analyticity and analytic continuation of Dirichlet-Neumann operators on doubly perturbed domains, J. Math. Fluid Mech., 10 (2008), 238-271. doi: 10.1007/s00021-006-0231-9.

[31]

T. Runst and W. Sickel, Sobolev Spaces of fractional order, Nemytskij operators and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, New York, 1996. doi: 10.1515/9783110812411.

[32]

A. Signorini, Questioni di elasticità non linearizzata e semilinearizzata, Rend. mat. e Appl, 18 (1959), 95-139.

[33]

J. J. Stoker, Water Waves: The Mathematical Theory with Applications. Pure and Applied Mathematics, Vol. IV., Interscience Publishers, Inc., New York, 1957.

[34]

J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal., 145 (1997), 136-150. doi: 10.1006/jfan.1996.3016.

[35]

H. Triebel, Theory of Function Spaces II, Birkhauser Verlag, 1992. doi: 10.1090/S0002-9939-2014-12243-X.

[36]

T. K. Wong, Blowup of solutions of the hydrostatic Euler equations, Proc. Amer. Math. Soc., 143 (2015), 1119-1125. doi: 10.1090/S0002-9939-2014-12243-X.

show all references

References:
[1]

P. W. Bates, On some nonlocal evolution equations arising in materials science, in: Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., 48 (2006), 13-52.

[2]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-16830-7.

[3]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66451-9.

[4]

P. Biler, G. Karch and W. A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 613-637. doi: 10.1016/S0294-1449(01)00080-4.

[5]

J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s \rightarrow 1$ and applications, J. Anal. Math., 87 (2002), 77-101. doi: 10.1007/BF02868470.

[6]

H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ. , 1 (2001), 387-404. doi: 10.1007/PL00001378.

[7]

L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119.

[8]

L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.

[9]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245.

[10]

C. Cao, S. Ibrahim, K. Nakanishi and E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337 (2015), 473-482. doi: 10.1007/s00220-015-2365-1.

[11]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, 2004. doi: 10.1201/9780203485217.

[12]

W. Craig and P. A. Worfolk, An integrable normal form for water waves in infinite depth, Phys. D, 84 (1995), 513-531. doi: 10.1016/0167-2789(95)00067-E.

[13]

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin-New York, 1976. doi: 10.1007/978-3-642-66165-5.

[14]

C. Fefferman and R. de la Llave, Relativistic stability of matter, I. Rev. Mat. Iberoamericana, 2 (1986), 119-213. doi: 10.4171/RMI/30.

[15]

C. L. Feffermana, D. S. McCormick, J. C. Robinsonb and J. L. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD and related models, J. Funct. Anal., 267 (2014), 1035-1056. doi: 10.1016/j.jfa.2014.03.021.

[16]

D. M. W. Frierson, A. J. Majda and O. M. Pauluis, Large scale dynamics of precipitation fronts in the tropical atmosphere: A novel relaxation limit, Commun. Math. Sci., 2 (2004), 591-626. doi: 10.4310/CMS.2004.v2.n4.a3.

[17]

Z. Jiang and Y. Zhou, Local existence for the generalized MHD equations,, preprint., (). 

[18]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0.

[19]

A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.

[20]

G. M. Kobelkov, Existence of a solution in the large for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. doi: 10.1016/j.crma.2006.04.020.

[21]

I. Kukavica, R. Temam, V. C. Vicol and M. Ziane, Existence and uniqueness of solutions for the hydrostatic Euler equations on a bounded domain with analytic data, C. R. Math. Acad. Sci. Paris, 348 (2010), 639-645. doi: 10.1016/j.crma.2010.03.023.

[22]

I. Kukavica, R. Temam, V. C. Vicol and M. Ziane, Local existence and uniqueness for the hydrostatic euler equations on a bounded domain, J. Differential Equations, 250 (2011), 1719-1746. doi: 10.1016/j.jde.2010.07.032.

[23]

J. Li and E. S. Titi, Global well-posedness of strong solutions to a tropical climate model,, preprint, (). 

[24]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equationsof the atmosphere and appliations, Nonlinearity, 5 (1992), 237-288.

[25]

J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002.

[26]

J. L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAO III), J. Math. Pures Appl., 74 (1995), 105-163.

[27]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511613203.

[28]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 77pp. doi: 10.1016/S0370-1573(00)00070-3.

[29]

C. Miao, J. Wu and Z. Zhang, Littlewood-Paley Theory and Its Applications in Partial Differential Equations of Fluid Dynamics (in Chinese), Science Press, Beijing, China, 2012.

[30]

D. P. Nicholls and M. Taber, Joint analyticity and analytic continuation of Dirichlet-Neumann operators on doubly perturbed domains, J. Math. Fluid Mech., 10 (2008), 238-271. doi: 10.1007/s00021-006-0231-9.

[31]

T. Runst and W. Sickel, Sobolev Spaces of fractional order, Nemytskij operators and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, New York, 1996. doi: 10.1515/9783110812411.

[32]

A. Signorini, Questioni di elasticità non linearizzata e semilinearizzata, Rend. mat. e Appl, 18 (1959), 95-139.

[33]

J. J. Stoker, Water Waves: The Mathematical Theory with Applications. Pure and Applied Mathematics, Vol. IV., Interscience Publishers, Inc., New York, 1957.

[34]

J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal., 145 (1997), 136-150. doi: 10.1006/jfan.1996.3016.

[35]

H. Triebel, Theory of Function Spaces II, Birkhauser Verlag, 1992. doi: 10.1090/S0002-9939-2014-12243-X.

[36]

T. K. Wong, Blowup of solutions of the hydrostatic Euler equations, Proc. Amer. Math. Soc., 143 (2015), 1119-1125. doi: 10.1090/S0002-9939-2014-12243-X.

[1]

Jinkai Li, Edriss Titi. Global well-posedness of strong solutions to a tropical climate model. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4495-4516. doi: 10.3934/dcds.2016.36.4495
[2]

Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203
[3]

Paola Goatin, Sheila Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Networks and Heterogeneous Media, 2016, 11 (1) : 107-121. doi: 10.3934/nhm.2016.11.107
[4]

Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure and Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605
[5]

Chaoying Li, Xiaojing Xu, Zhuan Ye. On long-time asymptotic behavior for solutions to 2D temperature-dependent tropical climate model. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1535-1568. doi: 10.3934/dcds.2021163
[6]

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
[7]

Young-Pil Choi, Seok-Bae Yun. A BGK kinetic model with local velocity alignment forces. Networks and Heterogeneous Media, 2020, 15 (3) : 389-404. doi: 10.3934/nhm.2020024
[8]

Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934/dcdsb.2020227
[9]

Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, 2021, 29 (6) : 3581-3607. doi: 10.3934/era.2021052
[10]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
[11]

Alessio Fiscella, Enzo Vitillaro. Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5015-5047. doi: 10.3934/dcds.2013.33.5015
[12]

Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297
[13]

Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2045-2063. doi: 10.3934/dcds.2017087
[14]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
[15]

Renhui Wan. Global well-posedness for the 2D Boussinesq equations with a velocity damping term. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2709-2730. doi: 10.3934/dcds.2019113
[16]

Elaine Cozzi, James P. Kelliher. Well-posedness of the 2D Euler equations when velocity grows at infinity. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2361-2392. doi: 10.3934/dcds.2019100
[17]

Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029
[18]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure and Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813
[19]

Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure and Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899
[20]

Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (204)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy