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

Hongjie Dong, Dong Li
2013 Transactions of the American Mathematical Society  
We consider a one-dimensional nonlocal nonlinear equation of the 1 2 is the fractional Laplacian and ν ≥ 0 is the viscosity coefficient. We primarily consider the regime 0 < α < 1 and 0 ≤ β ≤ 2 for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D α-patch models. In the critical and subcritical range 1 − α ≤ β ≤ 2, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range 0 ≤ β < 1 − α,
more » ... ange 0 ≤ β < 1 − α, 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.
doi:10.1090/s0002-9947-2013-06075-8 fatcat:63ltq7kxandqvkxbfwr56qbyle