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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[33]

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

[34]

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

[35]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[33]

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

[34]

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

[35]

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

[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.  Google Scholar

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