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- W2953092004 abstract "We consider a one-dimensional nonlocal nonlinear equation of the form: $partial_t u = (Lambda^{-alpha} u)partial_x u - nu Lambda^{beta}u$ where $Lambda =(-partial_{xx})^{frac 12}$ is the fractional Laplacian and $nuge 0$ is the viscosity coefficient. We consider primarily the regime $0<alpha<1$ and $0le beta le 2$ 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 $1-alphale beta le 2$, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range $0 le beta<1-alpha$, 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." @default.
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- W2953092004 date "2012-07-04" @default.
- W2953092004 modified "2023-09-26" @default.
- W2953092004 title "On a one-dimensional alpha-patch model with nonlocal drift and fractional dissipation" @default.
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- W2953092004 doi "https://doi.org/10.48550/arxiv.1207.0957" @default.
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