Abstract
We consider a one-dimensional nonlocal nonlinear equation of the form â t u = ( Î â Îą u ) â x u â ν Πβ u \partial _t u = (\Lambda ^{-\alpha } u)\partial _x u - \nu \Lambda ^{\beta }u , where Î = ( â â x x ) 1 2 \Lambda =(-\partial _{xx})^{\frac 12} is the fractional Laplacian and ν ⼠0 \nu \ge 0 is the viscosity coefficient. We primarily consider the regime 0 > Îą > 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 Îą \alpha -patch models. In the critical and subcritical range 1 â Îą ⤠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 â Îą 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.
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