  We consider a one-dimensional nonlocal nonlinear equation of the form:
$\partial_t u = (\Lambda^{-\alpha} u)\partial_x u - \nu \Lambda^{\beta}u$ where
$\Lambda =(-\partial_{xx})^{\frac 12}$ is the fractional Laplacian and $\nu\ge
0$ is the viscosity coefficient. We consider primarily the regime $0<\alpha<1$
and $0\le \beta \le 2$ for which the model has nonlocal drift, fractional
dissipation, and captures essential features of the 2D $\alpha$-patch models.
In the critical and subcritical range $1-\alpha\le \beta \le 2$, we prove
global wellposedness for arbitrarily large initial data in Sobolev spaces. In
the full supercritical range $0 \le \beta<1-\alpha$, we prove formation of
singularities in finite time for a class of smooth initial data. Our proof is
based on a novel nonlocal weighted inequality which can be of independent
interest.
