Dynamics of screened particles towards equi-spaced ground states

Abstract 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 {\mu-\mathrm{d}x} having zero average. To each of these measures μ we associate a (periodic) function u satisfying u ′ = d ⁢ x - μ {u^{\prime}=\mathrm{d}x-\mu} . For s ∈ ( 0 , 1 2 ) {s\in(0,\frac{1}{2})} we introduce energy functionals ℰ s ⁢ ( μ ) {{\mathscr{E}}^{s}(\mu)} that can be understood as the density of the s -Gagliardo seminorm of u per unit length. Since for s ≥ 1 2 {s\geq\frac{1}{2}} , the s -Gagliardo seminorms are infinite on functions with jumps, some regularization procedure is needed: For s ∈ [ 1 2 , 1 ) {s\in[\frac{1}{2},1)} we define ℰ ε s ⁢ ( μ ) := ℰ s ⁢ ( μ ε ) {{\mathscr{E}}_{\varepsilon}^{s}(\mu):={\mathscr{E}}^{s}(\mu_{\varepsilon})} , where μ ε