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

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 {\mu_{\varepsilon}}

is obtained by mollifying μ on scale ε. We prove that the minimizers of

ℰ s {{\mathscr{E}}^{s}}

and

ℰ ε s {{\mathscr{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 ∈ [ 1 2 , 1 ) {s\in[\frac{1}{2},1)}

the energy functionals

ℰ ε s {{\mathscr{E}}_{\varepsilon}^{s}}

blow up as

ε → 0 {\varepsilon\to 0}

, their gradients are uniformly bounded (with respect to ε), so that the corresponding trajectories converge, as

ε → 0 {\varepsilon\to 0}

, to the gradient flow solution of a suitable renormalized energy.