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

Abstract

We consider a one-dimensional nonlocal nonlinear equation of the form ∂tu = (Λ- αu)dxu-vΛ βu, where Λ = (-∂xx)1/2 is the fractional Laplacian and v > 0 is the viscosity coefficient. We primarily consider the regime 0 < α, < 1 and 0 < β < 2 for which the model has no local drift, fractional dissipation, and captures essential features of the 2D a-patch models. In the critical and sub critical 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, we prove formation of singularities in finite time for a class of smooth initial data. Our proof is based on a novel nonsocial weighted inequality which can be of independent interest. © 2013 American Mathematical Society Reverts to public domain 28 years from publication.