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 ∂ t u = ( Λ − α u ) ∂ x u − ν Λ β u \partial _t u = (\Lambda ^{-\alpha } u)\partial _x u - u \Lambda ^{\beta }u , where Λ = ( − ∂ x x ) 1 2 \Lambda =(-\partial _{xx})^{\frac 12} is the fractional Laplacian and ν ≥ 0 u \ge 0 is the viscosity coefficient. We primarily consider the regime 0 > α > 1 0>\alpha >1 and 0 ≤ β ≤ 2 0\le \beta \le 2 for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D α \alpha -patch models. In the critical and subcritical range 1 − α ≤ β ≤ 2 1-\alpha \le \beta \le 2 , we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range 0 ≤ β > 1 − α 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.