Global smoothness for a 1D supercritical transport model with nonlocal velocity

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).