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On a one-dimensional \(\alpha\)-patch model with nonlocal drift and fractional dissipation. (English) Zbl 1302.35334
Authors’ abstract: We consider a one-dimensional nonlocal nonlinear equation of the form \(\partial_tu=(\Lambda^{-\alpha}u)\partial_xu-\nu\Lambda^\beta u\), where \(\Lambda =(-\partial_{xx})^{\frac{1}{2}}\) is the fractional Laplacian and \(\nu\geq 0\) is the viscosity coefficient. We primarily consider the regime \(0<\alpha <1\) and \(0\leq\beta\leq 2\) 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 \(1-\alpha\leq\beta\leq 2\), we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range \(0\leq\beta <1-\alpha\), 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.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B44 Blow-up in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35R11 Fractional partial differential equations
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