# zbMATH — the first resource for mathematics

Remarks on the improved regularity criterion for the 2D Euler-Boussinesq equations with supercritical dissipation. (English) Zbl 1358.35139
Summary: This paper is devoted to the investigation of the regularity criterion to the two-dimensional (2D) Euler-Boussinesq equations with supercritical dissipation. By making use of the Littlewood-Paley technique, we provide an improved regularity criterion involving the temperature at the scaling invariant level, which improves the previous results.

##### 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 42B25 Maximal functions, Littlewood-Paley theory
Full Text:
##### References:
 [1] Adhikari, D; Cao, C; Shang, H; Wu, J; Xu, X; Ye, Z, Global regularity results for the 2D Boussinesq equations with partial dissipation, J. Differ. Equ., 260, 1893-1917, (2016) · Zbl 1328.35161 [2] Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, vol. 343. Springer, Berlin (2011) · Zbl 1227.35004 [3] 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 [4] Chae, D, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 497-513, (2006) · Zbl 1100.35084 [5] Chae, D; Nam, H, Local existence and blow-up criterion for the Boussinesq equations, Proc. R. Soc. Edinb. Sect. A, 127, 935-946, (1997) · Zbl 0882.35096 [6] 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 [7] Chen, Q; Miao, C; Zhang, Z, A new Bernstein inequality and the 2D dissipative quasigeostrophic equation, Commun. Math. Phys., 271, 821-838, (2007) · Zbl 1142.35069 [8] Constantin, P; Vicol, V, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22, 1289-1321, (2012) · Zbl 1256.35078 [9] Constantin, P; Wu, J, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincare Anal. Non Lineaire, 25, 1103-1110, (2008) · Zbl 1149.76052 [10] Constantin, P; Wu, J, Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 26, 159-180, (2009) · Zbl 1163.76010 [11] 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 [12] Danchin, R, Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics, Proc. Am. Math. Soc., 141, 1979-1993, (2013) · Zbl 1283.35080 [13] 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 [14] Weinan, E; Shu, C, Samll-scale structures in Boussinesq convection, Phys. Fluids, 6, 49-58, (1994) · Zbl 0822.76087 [15] Hmidi, T; Keraani, S; Rousset, F, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differ. Equ., 249, 2147-2174, (2010) · Zbl 1200.35228 [16] Hmidi, T; Keraani, S; Rousset, F, Global well-posedness for Euler-Boussinesq system with critical dissipation, Commun. Partial Differ. Equ., 36, 420-445, (2011) · Zbl 1284.76089 [17] Hmidi, T; Zerguine, M, On the global well-posedness of the Euler-Boussinesq system with fractional dissipation, Phys. D, 239, 1387-1401, (2010) · Zbl 1194.35329 [18] Hou, TY; Li, C, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 1-12, (2005) · Zbl 1274.76185 [19] Jiu, Q; Miao, C; Wu, J; Zhang, Z, The 2D incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46, 3426-3454, (2014) · Zbl 1319.35193 [20] 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 [21] Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2001) · Zbl 0983.76001 [22] Miao, C; Xue, L, On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, NoDEA Nonlinear Differ. Equ. Appl., 18, 707-735, (2011) · Zbl 1235.76020 [23] Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1987) · Zbl 0713.76005 [24] Qian, C, Blow-up criterion for the 2D Euler-Boussinesq system in terms of temperature, Electron. J. Differ. Equ., 2016, 1-11, (2016) · Zbl 1343.35196 [25] Silvestre, L, On the differentiablity of the solution to an equation with drift and fractional diffusion, Indiana Univ. Math. J., 61, 557-584, (2012) · Zbl 1308.35042 [26] Silvestre, L, Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11, 843-855, (2012) · Zbl 1263.35056 [27] Xu, X, Global regularity of solutions of 2D Boussinesq equations with fractional diffusion, Nonlinear Anal., 72, 677-681, (2010) · Zbl 1177.76024 [28] Xu, X; Xue, L, Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation, J. Differ. Equ., 256, 3179-3207, (2014) · Zbl 1452.76030 [29] Xu, X; Ye, Z, The lifespan of solutions to the inviscid 3D Boussinesq system, Appl. Math. Lett., 26, 854-859, (2013) · Zbl 1314.35113 [30] Xue, L., Ye, Z.: On the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion. arXiv:1601.03123 · Zbl 1379.35049 [31] Yang, W; Jiu, Q; Wu, J, Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation, J. Differ. Equ., 257, 4188-4213, (2014) · Zbl 1300.35108 [32] Ye, Z, Blow-up criterion of smooth solutions for the Boussinesq equations, Nonlinear Anal., 110, 97-103, (2014) · Zbl 1300.35109 [33] Ye, Z, On the regularity criterion for the 2D Boussinesq equations involving the temperature, Appl. Anal., 95, 615-626, (2016) · Zbl 1335.35207 [34] Ye, Z; Xu, X, Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation, J. Differ. Equ., 260, 6716-6744, (2016) · Zbl 1341.35135
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.