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Global regularity results for the 2D Boussinesq equations with vertical dissipation. (English) Zbl 1232.35111
Authors’ abstract: This paper furthers the study of the authors [J. Differ. Equations 249, No. 5, 1078–1088 (2010; Zbl 1193.35144)] on the global regularity issue concerning the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. It is shown here that the vertical velocity $$v$$ of any classical solution in the Lebesgue space $$L^q$$ with $$2\leq q<\infty$$ is bounded by $$C_1q$$ for $$C_1$$ independent of $$q$$. This bound significantly improves the previous exponential bound. In addition, we prove that, if $$v$$ satisfies $\int^T_0 \sup_{q\geq 2} {\| v(\cdot,t)\|^2_{L^q}\over q}\,dt< \infty,$ then the associated solution of the 2D Boussinesq equations preserves its smoothness on $$[0, T]$$. In particular, $$|v|_{L^q}\leq C_2\sqrt{q}$$ implies global regularity.

##### MSC:
 35Q30 Navier-Stokes equations 35B45 A priori estimates in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D09 Viscous-inviscid interaction 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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