  We are concerned with a nonlocal transport 1D-model with supercritical
dissipation $\gamma\in(0,1)$ in which the velocity is coupled via the Hilbert
transform, namely the so-called CCF model. This model arises as a lower
dimensional model for the famous 2D dissipative quasi-geostrophic equation and
in connection with vortex-sheet problems. It is known that its solutions can
blow up in finite time when $\gamma\in(0,1/2)$. On the other hand, as stated by
Kiselev (2010), in the supercritical subrange $\gamma\in\lbrack1/2,1)$ it is an
open problem to know whether its solutions are globally regular. We show global
existence of non-negative $H^{3/2}$-strong solutions in a supercritical
subrange (close to 1) that depends on the initial data norm. Then, for each
arbitrary smooth non-negative initial data, the model has a unique global
smooth solution provided that $\gamma\in\lbrack\gamma_{1},1)$ where
$\gamma_{1}$ depends on the $H^{3/2}$-initial data norm. Our approach is
inspired on that of Coti Zelati and Vicol (IUMJ, 2016).
