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The 2D Boussinesq equations with vertical viscosity and vertical diffusivity. (English) Zbl 1193.35144
Summary: This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the $$L^r$$-norm of the vertical velocity $$v$$ for any $$1<r<\infty$$ is globally bounded and that the $$L^{\infty }$$-norm of $$v$$ controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace $$(-\Delta )\delta$$ for $$\delta >0$$ would guarantee the global regularity of classical solutions.

##### 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 76R50 Diffusion
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