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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 $\nu\ge0$ 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, fractionaldissipation, and captures essential features of the 2D $\alpha$-patch models.In the critical and subcritical range $1-\alpha\le \beta \le 2$, we proveglobal wellposedness for arbitrarily large initial data in Sobolev spaces. Inthe full supercritical range $0 \le \beta<1-\alpha$, we prove formation ofsingularities in finite time for a class of smooth initial data. Our proof isbased on a novel nonlocal weighted inequality which can be of independentinterest.

On a one-dimensional \alpha-patch model with nonlocal drift and fractional dissipation