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

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

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.
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, 48 (2006), 13.

[2]

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

[3]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction,, Springer-Verlag, (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. 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. 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. doi: 10.1007/PL00001378.

[7]

L. A. Caffarelli, Further regularity for the Signorini problem,, Comm. Partial Differential Equations, 4 (1979), 1067. 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. 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. 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. doi: 10.1007/s00220-015-2365-1.

[11]

R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financial Mathematics Series, (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. doi: 10.1016/0167-2789(95)00067-E.

[13]

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics,, Springer-Verlag, (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. 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. 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. 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. 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. 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. 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. 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. 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.

[25]

J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean,, Nonlinearity, 5 (1992), 1007. 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.

[27]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (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). 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, (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. 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, (1996). doi: 10.1515/9783110812411.

[32]

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

[33]

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

[34]

J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations,, J. Funct. Anal., 145 (1997), 136. 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. 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, 48 (2006), 13.

[2]

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

[3]

J. Bergh and J. Löfström, Interpolation Spaces, An Introduction,, Springer-Verlag, (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. 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. 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. doi: 10.1007/PL00001378.

[7]

L. A. Caffarelli, Further regularity for the Signorini problem,, Comm. Partial Differential Equations, 4 (1979), 1067. 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. 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. 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. doi: 10.1007/s00220-015-2365-1.

[11]

R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financial Mathematics Series, (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. doi: 10.1016/0167-2789(95)00067-E.

[13]

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics,, Springer-Verlag, (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. 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. 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. 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. 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. 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. 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. 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. 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.

[25]

J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean,, Nonlinearity, 5 (1992), 1007. 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.

[27]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (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). 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, (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. 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, (1996). doi: 10.1515/9783110812411.

[32]

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

[33]

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

[34]

J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations,, J. Funct. Anal., 145 (1997), 136. 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. doi: 10.1090/S0002-9939-2014-12243-X.

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