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On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity
Global well-posedness of strong solutions to a tropical climate model
1. | Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel |
2. | Department of Mathematics, Texas A&M University, 3368-TAMU, College Station, TX 77843-3368, United States |
References:
[1] |
H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.
doi: 10.1016/0362-546X(80)90068-1. |
[2] |
H. Brézis and S. Wainger, A Note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.
doi: 10.1080/03605308008820154. |
[3] |
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. |
[4] |
C. Cao, J. Li and E. S. Titi, Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity, Arch. Rational Mech. Anal., 214 (2014), 35-76.
doi: 10.1007/s00205-014-0752-y. |
[5] |
C. Cao, J. Li and E. S. Titi, Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity, J. Differential Equations, 257 (2014), 4108-4132.
doi: 10.1016/j.jde.2014.08.003. |
[6] |
C. Cao, J. Li and E. S. Titi, Global well-posedness of the 3D primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl. Math., arXiv:1406.1995v1.
doi: 10.1002/cpa.21576. |
[7] |
C. Cao, J. Li and E. S. Titi, Strong solutions to the 3D primitive equations with horizontal dissipation: near $H^1$ initial data, preprint. |
[8] |
C. Cao, J. Li and E. S. Titi, Global well-posedness of the 3D primitive equations with horizontal viscosities and vertical diffusion, preprint. |
[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 and E. S. Titi, Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion, Comm. Math. Phys., 310 (2012), 537-568.
doi: 10.1007/s00220-011-1409-4. |
[11] |
R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635.
doi: 10.2307/1970954. |
[12] |
R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.
doi: 10.1090/S0002-9947-1975-0380244-8. |
[13] |
L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[14] |
E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8843-0. |
[15] |
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. |
[16] |
A. E. Gill, Some simple solutions for heat-induced tropical circulation, Quart. J. Roy. Meteor. Soc., 106 (1980), 447-462.
doi: 10.1002/qj.49710644905. |
[17] |
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. |
[18] |
I. Kukavica and M. Ziane, The regularity of solutions of the primitive equations of the ocean in space dimension three, C. R. Math. Acad. Sci. Paris, 345 (2007), 257-260.
doi: 10.1016/j.crma.2007.07.025. |
[19] |
I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753.
doi: 10.1088/0951-7715/20/12/001. |
[20] |
A. Larios, E. Lunasin and E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations, 255 (2013), 2636-2654.
doi: 10.1016/j.jde.2013.07.011. |
[21] |
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, arXiv:1502.06180.
doi: 10.1007/s00205-015-0946-y. |
[22] |
J. Li, E. S. Titi and Z. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb R^2$, Math. Models Methods Appl. Sci., 26 (2016), 803-822, arXiv:1410.1119.
doi: 10.1142/S0218202516500184. |
[23] |
J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.
doi: 10.1088/0951-7715/5/2/001. |
[24] |
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. |
[25] |
J. L. Lions, R. Temam and S. Wang, Mathematical theory for the coupled atmosphere-ocean models (CAO III), J. Math. Pures Appl., 74 (1995), 105-163. |
[26] |
A. J. Majda and J. A. Biello, The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J. Atmos. Sci., 60 (2003), 1809-1821.
doi: 10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2. |
[27] |
T. Matsuno, Quasi-geostrophic motions in the equatorial area, J. Meteor. Soc. Japan, 44 (1966), 25-42. |
[28] |
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] |
H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.
doi: 10.1016/0362-546X(80)90068-1. |
[2] |
H. Brézis and S. Wainger, A Note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.
doi: 10.1080/03605308008820154. |
[3] |
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. |
[4] |
C. Cao, J. Li and E. S. Titi, Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity, Arch. Rational Mech. Anal., 214 (2014), 35-76.
doi: 10.1007/s00205-014-0752-y. |
[5] |
C. Cao, J. Li and E. S. Titi, Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity, J. Differential Equations, 257 (2014), 4108-4132.
doi: 10.1016/j.jde.2014.08.003. |
[6] |
C. Cao, J. Li and E. S. Titi, Global well-posedness of the 3D primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl. Math., arXiv:1406.1995v1.
doi: 10.1002/cpa.21576. |
[7] |
C. Cao, J. Li and E. S. Titi, Strong solutions to the 3D primitive equations with horizontal dissipation: near $H^1$ initial data, preprint. |
[8] |
C. Cao, J. Li and E. S. Titi, Global well-posedness of the 3D primitive equations with horizontal viscosities and vertical diffusion, preprint. |
[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 and E. S. Titi, Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion, Comm. Math. Phys., 310 (2012), 537-568.
doi: 10.1007/s00220-011-1409-4. |
[11] |
R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635.
doi: 10.2307/1970954. |
[12] |
R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.
doi: 10.1090/S0002-9947-1975-0380244-8. |
[13] |
L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[14] |
E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8843-0. |
[15] |
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. |
[16] |
A. E. Gill, Some simple solutions for heat-induced tropical circulation, Quart. J. Roy. Meteor. Soc., 106 (1980), 447-462.
doi: 10.1002/qj.49710644905. |
[17] |
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. |
[18] |
I. Kukavica and M. Ziane, The regularity of solutions of the primitive equations of the ocean in space dimension three, C. R. Math. Acad. Sci. Paris, 345 (2007), 257-260.
doi: 10.1016/j.crma.2007.07.025. |
[19] |
I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753.
doi: 10.1088/0951-7715/20/12/001. |
[20] |
A. Larios, E. Lunasin and E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations, 255 (2013), 2636-2654.
doi: 10.1016/j.jde.2013.07.011. |
[21] |
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, arXiv:1502.06180.
doi: 10.1007/s00205-015-0946-y. |
[22] |
J. Li, E. S. Titi and Z. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb R^2$, Math. Models Methods Appl. Sci., 26 (2016), 803-822, arXiv:1410.1119.
doi: 10.1142/S0218202516500184. |
[23] |
J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.
doi: 10.1088/0951-7715/5/2/001. |
[24] |
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. |
[25] |
J. L. Lions, R. Temam and S. Wang, Mathematical theory for the coupled atmosphere-ocean models (CAO III), J. Math. Pures Appl., 74 (1995), 105-163. |
[26] |
A. J. Majda and J. A. Biello, The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J. Atmos. Sci., 60 (2003), 1809-1821.
doi: 10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2. |
[27] |
T. Matsuno, Quasi-geostrophic motions in the equatorial area, J. Meteor. Soc. Japan, 44 (1966), 25-42. |
[28] |
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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