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 μ on the real line that are screened by the Lebesgue measure, i.e., with μ-d x μ-dx having zero average. To each of these measures μ we associate a (periodic) function u satisfying u ′ = d x-μ {u′=dx-μ. For s (0, 1 2) {e(0, 1{2) we introduce energy functionals s (μ) Es(μ) that can be understood as the density of the s-Gagliardo seminorm of u per unit length. Since for s ≥ 1 2 {s1/2, the s-Gagliardo seminorms are infinite on functions with jumps, some regularization procedure is needed: For s [ 1 2, 1) e1{2,1) we define ϵ s (μ):= s (μ ϵ) Es(μ):= Es(μ), where μ ϵ μ is obtained by mollifying μ on scale ϵ. We prove that the minimizers of s Es and ϵ s Es 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 [ 1 2, 1) e1/2,1) the energy functionals ϵ s Es blow up as ϵ → 0 to 0, their gradients are uniformly bounded (with respect to ϵ), so that the corresponding trajectories converge, as ϵ → 0 to 0, to the gradient flow solution of a suitable renormalized energy.