2025/05/27 by De Luca, Lucia, Goldman, Michael, Ponsiglione, Marcello · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2505.21768
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 having zero average. To each of these measures μ we associate a (periodic) function u satisfying u'= d x - μ. For s∈ (0,\frac 12) we introduce energy functionals \mathcal Es(μ) that can be understood as the density of the s-Gagliardo seminorm of u per unit length. Since for s≥ \frac 12, the s-Gagliardo seminorms are infinite on functions with jumps, some regularization procedure is needed: For s∈[\frac 12,1) we define \mathcal E_\es(μ):= \mathcal Es(μ_\e), where με is obtained by mollifying μ on scale ε. We prove that the minimizers of \mathcal Es and \mathcal Eεs 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∈[\frac 12 ,1) the energy functionals \mathcal Eεs blow up as ε→ 0, their gradients are uniformly bounded (with respect to ε), so that the corresponding trajectories converge, as ε→ 0, to the gradient flow solution of a suitable renormalized energy.