Blow-up criterion of solutions of the horizontal viscous primitive equations with horizontal eddy diffusivity. (English) Zbl 1529.35523
Summary: In this paper, we consider the blow-up criterion of the \(H^1\) norm of solutions to the primitive equations with only horizontal viscosity and horizontal thermal diffusivity. The maximum norm of the gradient of the vorticity is showed to control the break down of the \(H^1\) norm of solutions when there is no additional symmetry assumption for the data. The temperature irrelevant to the break down is also showed.
MSC:
| 35Q86 | PDEs in connection with geophysics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 86A05 | Hydrology, hydrography, oceanography |
| 86A10 | Meteorology and atmospheric physics |
| 76U60 | Geophysical flows |
| 35B44 | Blow-up in context of PDEs |
References:
| [1] | Pedlosky, J., Geophysical Fluid Dynamics (1987), Springer-Verlag: Springer-Verlag New York · Zbl 0713.76005 |
| [2] | Temam, R.; Ziane, M., Some mathematical problems in geophysical fluid dynamics, (Handbook of mathematical fluid dynamics, Vol. III (2004), North-Holland: North-Holland Amsterdam), 535-657 · Zbl 1222.35145 |
| [3] | Guo, B. L.; Huang, D. W., Infinite-Dimensional Dynamical Systems in Atmospheric and Oceanic Science (2014), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ; Zhejiang Science and Technology Publishing House, Hangzhou · Zbl 1303.86002 |
| [4] | Majda, A. J., (Introduction to PDEs and Waves for the Atmosphere and Ocean. Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, Vol. 9 (2003), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1278.76004 |
| [5] | Salmon, R., Lectures on Geophysical Fluid Dynamics (1998), Oxford Univ. Press: Oxford Univ. Press New York |
| [6] | Zeng, Q. C., Mathematical and Physical Foundations of Numerical Weather Prediction (1979), Science Press: Science Press Beijing, (in Chinese) |
| [7] | Lions, J. L.; Temam, R.; Wang, S., New formulations of the primitive equations of the atmosphere and appliations, Nonlinearity, 5, 237-288 (1992) · Zbl 0746.76019 |
| [8] | Lions, J. L.; Temam, R.; Wang, S., On the equations of the large-scale ocean, Nonlinearity, 5, 1007-1053 (1992) · Zbl 0766.35039 |
| [9] | Lions, J. L.; Temam, R.; Wang, S., Mathematical study of the coupled models of atmosphere and ocean (CAO III), J. Math. Pures Appl., 74, 105-163 (1995) · Zbl 0866.76025 |
| [10] | Bresch, D.; Guillén-González, F.; Masmoudi, N.; Rodríguez-Bellido, M. A., On the uniqueness of weak solutions of the two-dimensional primitive equations, Differential Integral Equations, 16, 77-94 (2003) · Zbl 1042.35042 |
| [11] | Petcu, M.; Temam, R.; Ziane, M., Some mathematical problems in geophysical fluid dynamics, (Handbook of Numerical Analysis, Vol. 14 (2009), Elsevier), 577-750 |
| [12] | Tachim Medjo, T., On the uniqueness of \(z\)-weak solutions of the threedimensional primitive equations of the ocean, Nonlinear Anal. RWA, 11, 1413-1421 (2010) · Zbl 1189.35005 |
| [13] | Kukavica, I.; Pei, Y.; Rusin, W.; Ziane, M., Primitive equations with continuous initial data, Nonlinearity, 27, 1135-1155 (2014) · Zbl 1291.35187 |
| [14] | Li, J. K.; Titi, E. S., Existence and uniqueness of weak solutions to viscous primitive equations for a certain class of discontinuous initial data, SIAM J. Math. Anal., 49, 1, 1-28 (2017) · Zbl 1364.35275 |
| [15] | Guillén-González, F.; Masmoudi, N.; Rodríguez-Bellido, M. A., Anisotropic estimates and strong solutions of the primitive equations, Differential Integral Equations, 14, 1381-1408 (2001) · Zbl 1161.76454 |
| [16] | Bresch, D.; Kazhikhov, A.; Lemoine, J., On the two-dimensional hydrostatic Navier-Stokes equations, SIAM J. Math. Anal., 36, 3, 796-814 (2004), (electronic) · Zbl 1075.35034 |
| [17] | Cao, C. S.; Titi, E. S., Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. Math., 166, 245-267 (2007) · Zbl 1151.35074 |
| [18] | Kobelkov, G. M., Existence of a solution ‘in the large’ for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343, 4, 283-286 (2006) · Zbl 1102.35003 |
| [19] | Guo, B. L.; Huang, D. W., 3D stochastic primitive equations of the large-scale ocean: Global well-posedness and attractors, Commun. Math. Phys., 286, 697-723 (2009) · Zbl 1179.37115 |
| [20] | Guo, B. L.; Huang, D. W., Existence of the universal attractor for the 3-D viscous primitive equations of large-scale moist atmosphere, J. Differential Equations, 251, 457-491 (2011) · Zbl 1229.35181 |
| [21] | Aźerad, P.; Guillén, F., Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics, SIAM J. Math. Anal., 33, 847-859 (2001) · Zbl 0999.35072 |
| [22] | Li, J. K.; Titi, E. S., The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: Rigorous justification of the hydrostatic approximation, J. Math. Pures Appl., 124, 9, 30-58 (2019) · Zbl 1412.35224 |
| [23] | Furukawa, K.; Giga, Y.; Hieber, M.; Hussein, A.; Kashiwabara, T.; Wrona, M., Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier-Stokes equations, Nonlinearity, 33, 12, 6502-6516 (2020) · Zbl 1452.35150 |
| [24] | Liu, X.; Titi, E. S., Global existence of weak solutions to the compressible primitive equations of atmospheric dynamics with degenerate viscosities, SIAM J. Math. Anal., 51, 1913-1964 (2019) · Zbl 1419.35150 |
| [25] | Liu, X.; Titi, E. S., Local well-posedness of strong solutions to the three-dimensional compressible primitive equations, Arch. Ration. Mech. Anal., 241, 729-764 (2021) · Zbl 1472.35393 |
| [26] | Wang, F. C.; Jiu, Q. S.; Xu, X. J., Global well-posedness of strong solutions to the 2D nonhomogeneous incompressible primitive equations with vacuum, J. Differential Equations, 286, 624-644 (2021) · Zbl 1466.35347 |
| [27] | Lian, R. X.; Zeng, Q. C., Existence of a strong solution and trajectory attractor for a climate dynamics model with topography effffects, J. Math. Anal. Appl., 458, 628-675 (2018) · Zbl 1375.35561 |
| [28] | Cao, C. S.; Ibrahim, S.; Nakanishi, K.; Titi, E. S., Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337, 2, 473-482 (2015) · Zbl 1317.35262 |
| [29] | Wong, T. K., Blowup of solutions of the hydrostatic Euler equations, Proc. Amer. Math. Soc., 143, 3, 1119-1125 (2015) · Zbl 1311.35200 |
| [30] | Cao, C. S.; Li, J. K.; Titi, E. S., Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity, Arch. Ration. Mech. Anal., 214, 35-76 (2014) · Zbl 1305.35113 |
| [31] | Cao, C. S.; Li, J. K.; Titi, E. S., Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity, J. Differential Equations, 257, 4108-4132 (2014) · Zbl 1300.35089 |
| [32] | Cao, C. S.; Li, J. K.; Titi, E. S., Global well-posedness of the 3D primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl. Math., 69, 8, 1492-1531 (2016) · Zbl 1351.35125 |
| [33] | Cao, C. S.; Li, J. K.; Titi, E. S., Strong solutions to the 3D primitive equations with horizontal dissipation: near \(H^1\) initial data, J. Funct. Anal., 272, 11, 4606-4641 (2017) · Zbl 1366.35123 |
| [34] | Cao, C. S.; Li, J. K.; Titi, E. S., Global well-posedness of the 3D primitive equations with horizontal viscosities and vertical diffusion, Physica D, 412, Article 132606 pp. (2020), 25pp · Zbl 1484.35315 |
| [35] | Cao, C. S.; Titi, E. S., Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion, Comm. Math. Phys., 310, 537-568 (2012) · Zbl 1267.35157 |
| [36] | Beale, J. T.; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commn. Math. Phys., 94, 61-66 (1984) · Zbl 0573.76029 |
| [37] | Ishimura, N.; Morimoto, H., Remarks on the blow-up criterion for the 3-D Boussinesq equations, Math. Models Methods Appl. Sci., 9, 9, 1323-1332 (1999) · Zbl 1034.35113 |
| [38] | Saffman, P. G., Vortex Dynamics (1992), Cambridge Univ. Press: Cambridge Univ. Press New York · Zbl 0777.76004 |
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