Dynamics of screened particles towards equi-spaced ground states

This paper deals with the dynamics - driven by the gradient flow of negative fractional seminorms - of empirical measures towards equi-spaced ground states. Specifically, we consider periodic empirical measures $\mu$ on the real line that are screened by the Lebesgue measure, i.e., with $\mu-d x$ having zero average. To each of these measures $\mu$ we associate a {(periodic)} function $u$ satisfying $u'= d x - \mu$. For $s\in (0,\frac 12)$ we introduce energy functionals $\mathcal E^s(\mu)$ that can be understood as the density of the $s$-Gagliardo seminorm of $u$ per unit length. Since for $s\ge \frac 12$, the $s$-Gagliardo seminorms are infinite on functions with jumps, some regularization procedure is needed: For $s\in[\frac 12,1)$ we define $\mathcal E_\e^s(\mu):= \mathcal E^s(\mu_\e)$, where $\mu_\varepsilon$ is obtained by mollifying $\mu$ on scale $\varepsilon$. We prove that the minimizers of $\mathcal E^s$ and $\mathcal E_\varepsilon^s$ are the equi-spaced configurations of particles with lattice spacing equal to one. Then, we prove the exponential convergence of the corresponding gradient flows to the equi-spaced steady states. Finally, although for $s\in[\frac 12 ,1)$ the energy functionals $\mathcal E_\varepsilon^s$ blow up as $\varepsilon\to 0$, their gradients are uniformly bounded (with respect to $\varepsilon$), so that the corresponding trajectories converge, as $\varepsilon\to 0$, to the gradient flow solution of a suitable renormalized energy.