Calculus of Variations and Geometric Measure Theory

J. A. Carrillo - S. Lisini - E. Mainini

Gradient flows for non-smooth interaction potentials

created by lisini on 20 Jun 2012
modified by mainini on 27 Feb 2014


Published Paper

Inserted: 20 jun 2012
Last Updated: 27 feb 2014

Journal: Nonlinear Anal. TMA
Volume: 100
Pages: 122-147
Year: 2014


We deal with a nonlocal interaction equation describing the evolution of a particle density under the effect of a general symmetric pairwise interaction potential, not necessarily in convolution form. We describe the case of a convex (or $\lambda$-convex) potential, possibly not smooth at several points, generalizing the results of CDFLS. We also identify the cases in which the dynamic is still governed by the continuity equation with well-characterized nonlocal velocity field.

Reference: CDFLS J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, D. Slepcev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J. 156 (2011), 229--271.

Keywords: Wasserstein distance, Gradient flows, aggregation equations , measure solution