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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 |
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. |
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