Global regularity for the 2D anisotropic Boussinesq Equations with vertical dissipation

By Chongsheng Cao, Jiahong Wu. 2011

91 citations recorded by the source corpus.

This paper establishes the global in time existence of classical solutions to the 2D anisotropic Boussinesq equations with vertical dissipation. When only the vertical dissipation is present, there is no direct control on the horizontal derivatives and the global regularity problem is very challenging. To solve this problem, we bound the derivatives in terms of the $L^\infty$-norm of the vertical velocity $v$ and prove that $\|v\|_{L^{r}}$ with $2\le r<\infty$ at any time does not grow faster than $\sqrt{r \log r}$ as $r$ increases. A delicate interpolation inequality connecting $\|v\|_{L^\infty}$ and $\|v\|_{L^r}$ then yields the desired global regularity.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/977015d86545a4a0614f89a68061b405bc769fb4

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion (2013, methodology): “2 does not directly follow from the methods in [2] on the 2D Boussinesq equations.”
Global regularity results for the 2D Boussinesq equations with vertical dissipation (2011, unknown): “Through careful energy estimates and applying anisotropic inequalities for triple-product terms, [5] observe that, if”