  We investigate a class of aggregation-diffusion equations with strongly
singular kernels and weak (fractional) dissipation in the presence of an
incompressible flow. Without the flow the equations are supercritical in the
sense that the tendency to concentrate dominates the strength of diffusion and
solutions emanating from sufficiently localised initial data may explode in
finite time. The main purpose of this paper is to show that under suitable
spectral conditions on the flow, which guarantee good mixing properties, for
any regular initial datum the solution to the corresponding
advection-aggregation-diffusion equation is global if the prescribed flow is
sufficiently fast. This paper can be seen as a partial extension of Kiselev and
Xu (Arch. Rat. Mech. Anal. 222(2), 2016), and our arguments show in particular
that the suppression mechanism for the classical 2D parabolic-elliptic
Keller-Segel model devised by Kiselev and Xu also applies to the fractional
Keller-Segel model (where $\triangle$ is replaced by $-\Lambda^\gamma$)
requiring only that $\gamma>1$. In addition, we remove the restriction to
dimension $d<4$.
