Abstract
We consider an aggregation equation in $\mathbb R^d$, $d\ge 2$ with fractional dissipation: $u_t + \nabla\cdot(u \nabla K*u)=-\nu \Lambda^\gamma u $, where $\nu\ge 0$, $0 < \gamma\le 2$ and $K(x)=e^{-|x|}$. In the supercritical case, $0 < \gamma < 1$, we obtain new local wellposedness results and smoothing properties of solutions. In the critical case, $\gamma=1$, we prove the global wellposedness for initial data having a small $L_x^1$ norm. In the subcritical case, $\gamma > 1$, we prove global wellposedness and smoothing of solutions with general $L_x^1$ initial data.
Citation
Dong Li . José L. Rodrigo . "Wellposedness and regularity of solutions of an aggregation equation." Rev. Mat. Iberoamericana 26 (1) 261 - 294, March, 2010.
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