Revista Matemática Iberoamericana


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Volume 26, Issue 1, 2010, pp. 261–294
DOI: 10.4171/RMI/601

Published online: 2010-04-30

Wellposedness and regularity of solutions of an aggregation equation

Dong Li[1] and José L. Rodrigo[2]

(1) University of Iowa, Iowa City, United States
(2) University of Warwick, Coventry, UK

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.

Keywords: Aggregation equations, well-posedness, higher regularity

Li Dong, Rodrigo José: Wellposedness and regularity of solutions of an aggregation equation. Rev. Mat. Iberoamericana 26 (2010), 261-294. doi: 10.4171/RMI/601