Vanishing viscosity limit for aggregation-diffusion equations

This article is devoted to the convergence analysis of the diffusive approximation of the measure-valued solutions to the so-called aggregation equation, which is now widely used to model collective motion of a population directed by an interaction potential. We prove, over the whole space in any dimension, a uniform-in-time convergence in Wasserstein distance in all finite-time intervals, in the general framework of Lipschitz continuous potentials, and provide an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msqrt> <mml:mi>ε</mml:mi> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> rate, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> is the diffusion parameter, when the potential is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> -convex. We give an extension to some repulsive potentials and prove sharp convergence rates of the steady states towards the Dirac mass, under some uniform attractiveness assumptions.