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 Rd\mathbb R^d, d2d\ge 2 with fractional dissipation: ut+(uKu)=νΛγuu_t + \nabla\cdot(u \nabla K*u)=-\nu \Lambda^\gamma u , where ν0\nu\ge 0, 0<γ20 < \gamma\le 2 and K(x)=exK(x)=e^{-|x|}. In the supercritical case, 0<γ<10 < \gamma < 1, we obtain new local wellposedness results and smoothing properties of solutions. In the critical case, γ=1\gamma=1, we prove the global wellposedness for initial data having a small Lx1L_x^1 norm. In the subcritical case, γ>1\gamma > 1, we prove global wellposedness and smoothing of solutions with general Lx1L_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