We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and one as the prefactor of the fractional Laplacian. We obtain existence and convergence results in certain parameter regimes and limits. We study the long time behaviour of solutions to the general problem and prove the existence of a unique global attractor. We apply the results to two particular active scalar equations arising in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.

我们在分数拉普拉斯算子的背景下研究对流扩散方程的抽象族的性质。有两个独立的扩散参数进入系统，一个通过漂移速度的本构定律，另一个是分数拉普拉斯算子的前置因子。我们在某些参数范围和限制条件下获得了存在和收敛的结果。我们研究了解决一般问题的长期行为，并证明了独特的全球吸引子的存在。我们将结果应用到地球物理流体动力学中产生的两个特殊的活动标量方程，即表面准地转方程和磁地转方程。