×

zbMATH — the first resource for mathematics

Blow-up criterion of smooth solutions for the Boussinesq equations. (English) Zbl 1300.35109
Summary: This paper is dedicated to the study the Boussinesq equations with fractional dissipation in \(n\)-dimensions (\(n \geq 2\)). We obtain a blow-up criterion for the local in time classical solution to the Boussinesq equations with arbitrarily small fractional powers of the Laplacian. Therefore our results significantly improve the previous works and are more closer to the resolution of the well-known regularity criterion issue on the inviscid Boussinesq equations.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B44 Blow-up in context of PDEs
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Constantin, P.; Doering, C. R., Infinite Prandtl number convection, J. Stat. Phys., 94, 159-172, (1999) · Zbl 0935.76083
[2] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer-Verlag New York · Zbl 0713.76005
[3] Majda, A.; Bertozzi, A., Vorticity and incompressible flow, (2001), Cambridge University Press Cambridge
[4] Danchin, R.; Paicu, M., Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci., 21, 421-457, (2011) · Zbl 1223.35249
[5] Cannon, J.; DiBenedetto, E., (The Initial Value Problem for the Boussinesq Equations with Data in \(L^p\), Lecture Notes in Mathematics, vol. 771, (1980), Springer Berlin), 129-144
[6] Cao, C.; Wu, J., Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech., 208, 985-1004, (2013) · Zbl 1284.35140
[7] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 497-513, (2006) · Zbl 1100.35084
[8] Chae, D.; Wu, J., The 2D Boussinesq equations with logarithmically supercritical velocities, Adv. Math., 230, 1618-1645, (2012) · Zbl 1248.35156
[9] Shu, W. E, C., Samll-scale structures in Boussinesq convection, Phys. Fluids, 6, 49-58, (1994) · Zbl 0822.76087
[10] Hou, T. Y.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 1-12, (2005) · Zbl 1274.76185
[11] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations, 249, 2147-2174, (2010) · Zbl 1200.35228
[12] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36, 420-445, (2011) · Zbl 1284.76089
[13] Larios, A.; Lunasin, E.; Titi, E. S., Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations, 255, 2636-2654, (2013) · Zbl 1284.35343
[14] Jiu, Q.; Miao, C.; Wu, J.; Zhang, Z., The 2D incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., (2014), in press
[15] Miao, C.; Xue, L., On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differential Equations Appl., 18, 707-735, (2011) · Zbl 1235.76020
[16] Xu, X., Global regularity of solutions of 2D Boussinesq equations with fractional diffusion, Nonlinear Anal., 72, 677-681, (2010) · Zbl 1177.76024
[17] Chae, D.; Nam, H. S., Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A, 127, 935-946, (1997) · Zbl 0882.35096
[18] Cui, X.; Dou, C.; Jiu, Q., Local well-posedness and blow up criterion for the inviscid Boussinesq system in Hölder spaces, J. Partial Differ. Equ., 25, 220-238, (2012) · Zbl 1274.35280
[19] Chae, D.; Kim, S.; Nam, H., Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations, Nagoya Math. J., 155, 55-80, (1999) · Zbl 0939.35150
[20] Danchin, R., Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics, Proc. Amer. Math. Soc., 141, 1979-1993, (2013) · Zbl 1283.35080
[21] Ishimura, N.; Morimoto, H., Remarks on the blow-up criterion for the 3D Boussinesq equations, Math. Methods Appl. Sci., 9, 1323-1332, (1999) · Zbl 1034.35113
[22] Liu, X.; Wang, M.; Zhang, Z., Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces, J. Math. Fluid Mech., 12, 280-292, (2010) · Zbl 1195.76136
[23] Xu, X.; Ye, Z., The lifespan of solutions to the inviscid 3D Boussinesq system, Appl. Math. Lett., 26, 854-859, (2013) · Zbl 1314.35113
[24] Taniuchi, Y., A note on the blow-up criterion for the inviscid 2D Boussinesq equations, (Salvi, R., The Navier-Stokes Equations: Theory and Numerical Methods, Lecture Notes in Pure and Applied Mathematics, vol. 223, (2002)), 131-140 · Zbl 0991.35070
[25] Chae, D., Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal., 38, 3-4, 339-358, (2004), A printed version of RIM-GARC (Seoul National University, Korea) preprint no. 8 (June, 2001) · Zbl 1068.35097
[26] Qiu, H.; Du, Y.; Yao, Z., A blow-up criterion for 3D Boussinesq equations in Besov spaces, Nonlinear Anal., 73, 806-815, (2010) · Zbl 1193.76025
[27] Qiu, H.; Du, Y.; Yao, Z., Local existence and blow-up criterion for the generalized Boussinesq equations in Besov spaces, Math. Methods Appl. Sci., 36, 86-98, (2013) · Zbl 1257.35153
[28] Beale, J. T.; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 94, 61-66, (1984) · Zbl 0573.76029
[29] Kozono, H.; Ogawa, T.; Taniuchi, Y., The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242, 251-278, (2002) · Zbl 1055.35087
[30] Fan, J.; Ozawa, T., Regularity criteria for the 3D density-dependent Boussinesq equations, Nonlinearity, 22, 553-568, (2009) · Zbl 1168.35416
[31] Fan, J.; Zhou, Y., A note on regularity criterion for the 3D Boussinesq system with partial viscosity, Appl. Math. Lett., 22, 802-805, (2009) · Zbl 1171.35342
[32] Ye, Z., A note on global well-posedness of solutions to Boussinesq equations with fractional dissipation, Acta Math. Sci., (2014), in press
[33] Kato, T., (Liapunov Functions and Monotonicity in the Euler and Navier-Stokes Equations, Lecture Notes in Mathematics, vol. 1450, (1990), Springer-Verlag Berlin)
[34] Córdoba, A.; Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249, 3, 511-528, (2004) · Zbl 1309.76026
[35] Lei, Z.; Zhou, Y., BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25, 575-583, (2009) · Zbl 1171.35452
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.