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

Abstract

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 $\Vert v{\Vert }_{L^r}$ 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 $\Vert v{\Vert }_{L^{\infty }}$ and $\Vert v{\Vert }_{L^r}$ then yields the desired global regularity.

17 pages

References in corpus (1)

• Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces

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• A global regularity result for the 2D Boussinesq equations with critical dissipation
• Nonlinear maximum principles for dissipative linear nonlocal operators and applications
• Global smooth solution to the 2D Boussinesq equations with fractional dissipation
• Small global solutions to the damped two-dimensional Boussinesq equations
• Hölder Continuous Periodic Solution of Boussinesq Equation with Partial Viscosity
• The 2D Boussinesq equations with logarithmically supercritical velocities