Boussinesq Equations with Partial or Fractional Dissipation

Abstract

The two-dimensional (2D) incompressible Boussinesq system is not only an important model in geophysics, but also retains some key features of the 3D Euler and Navier-Stokes equations such as the vortex stretching mechanism. Especially, the inviscid 2D Boussinesq equations are identical to the Euler equations for the 3D axisymmetric swirling flows. Even though the global regularity of full dissipative Boussinesq equations is well known, the global regularity problem of inviscid case is still left open. First, we prove the global existence and uniqueness of 2D Boussinesq equations with partial dissipation in bounded main with Navier type boundary conditions. Secondly, we investigate Boussinesq equations with fractional dissipation on a d-dimensional periodic domain, and apply a re-developed tool of LittlewoodPaley decomposition to achieve global existence and uniqueness of weak solutions. Lastly, we focus on several variants of the 2D incompressible Euler equations. It is not known whether global well-posedness result would hold if there is only partially damping term for 2D Euler equation. Besides, in the vorticity equations, the partially damping term becomes a non-local operator \mathcal R_2^2 \omega. Our numerical simulations show that by replacing \mathcal R_2^2 \omega with different operators (e.g. \mathcal R_1\mathcal R_2 \omega), the solutions will behave quite differently.