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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.21768 |
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| _version_ | 1866908382005297152 |
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| author | De Luca, Lucia Goldman, Michael Ponsiglione, Marcello |
| author_facet | De Luca, Lucia Goldman, Michael Ponsiglione, Marcello |
| contents | 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 $s\in (0,\frac 12)$ we introduce energy functionals $\mathcal E^s(μ)$ 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 $\mathcal E_\e^s(μ):= \mathcal E^s(μ_\e)$, where $μ_\varepsilon$ is obtained by mollifying $μ$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_21768 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamics of screened particles towards equi-spaced ground states De Luca, Lucia Goldman, Michael Ponsiglione, Marcello Functional Analysis 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 $s\in (0,\frac 12)$ we introduce energy functionals $\mathcal E^s(μ)$ 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 $\mathcal E_\e^s(μ):= \mathcal E^s(μ_\e)$, where $μ_\varepsilon$ is obtained by mollifying $μ$ 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. |
| title | Dynamics of screened particles towards equi-spaced ground states |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2505.21768 |