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Remarks on global regularity of the 2D Boussinesq equations with fractional dissipation. (English) Zbl 1405.35171
Summary: In this paper, we are interested in the study of the Cauchy problem to the two-dimensional (2D) incompressible Boussinesq equations with fractional dissipation. By making use of the nonlinear lower bounds for the fractional Laplacian established in [P. Constantin and V. Vicol, Geom. Funct. Anal. 22, No. 5, 1289–1321 (2012; Zbl 1256.35078)], we establish the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. This result significantly improves the recent works of Constantin and Vicol [loc. cit.] and W. Yang et al. [J. Differ. Equations 257, No. 11, 4188–4213 (2014; Zbl 1300.35108)].

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
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