# Wellposedness and regularity of solutions of an aggregation equation

### Dong Li

University of Iowa, Iowa City, United States### José L. Rodrigo

University of Warwick, Coventry, UK

## 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.

## Cite this article

Dong Li, José L. Rodrigo, Wellposedness and regularity of solutions of an aggregation equation. Rev. Mat. Iberoam. 26 (2010), no. 1, pp. 261–294

DOI 10.4171/RMI/601