Asymptotic Behavior of Gradient Flows Driven by Nonlocal Power Repulsion and Attraction Potentials in One Dimension

We study the long time behavior of the Wasserstein gradient flow for an energy functional consisting of two components: particles are attracted to a fixed profile $\omega$ by means of an interaction kernel $\psi_a(z)=|z|^{q_a}$,and they repel each other by means of another kernel $\psi_r(z)=|z|^{q_r}$. We focus on the case of one space dimension and assume that $1\le q_r\le q_a\le 2$. Our main result is that the flow converges to an equilibrium if either $q_r<q_a$ or $1\le q_r=q_a\le4/3$,and if the solution has the same (conserved) mass as the reference state $\omega$. In the cases $q_r=1$ and $q_r=2$, we are able to discuss the behavior for different masses as well, and we explicitly identify the equilibrium state, which is independent of the initial condition. Our proofs heavily use the inverse distribution function of the solution.