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

Abstract:

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 primarily consider 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.