Wellposedness and regularity of solutions of an aggregation equation

Abstract

We consider an aggregation equation in ${\mathbb{R}}^d$, $d\ge 2$ with fractional dissipation: $u_t+\nabla \cdot (u\nabla K\ast u)=-\nu {\Lambda }^{\gamma }u$ , 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.