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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>μ</m:mi> <m:mo>-</m:mo> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mo>⁢</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:math> {\mu-\mathrm{d}x} having zero average. To each of these measures μ we associate a (periodic) function u satisfying <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>u</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mo>⁢</m:mo> <m:mi>x</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mi>μ</m:mi> </m:mrow> </m:mrow> </m:math> {u^{\prime}=\mathrm{d}x-\mu} . For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {s\in(0,\frac{1}{2})} we introduce energy functionals <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="script">ℰ</m:mi> <m:mi>s</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>μ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{\mathscr{E}}^{s}(\mu)} that can be understood as the density of the s -Gagliardo seminorm of u per unit length. Since for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>s</m:mi> <m:mo>≥</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:math> {s\geq\frac{1}{2}} , the s -Gagliardo seminorms are infinite on functions with jumps, some regularization procedure is needed: For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {s\in[\frac{1}{2},1)} we define <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mi mathvariant="script">ℰ</m:mi> <m:mi>ε</m:mi> <m:mi>s</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>μ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>:=</m:mo> <m:mrow> <m:msup> <m:mi mathvariant="script">ℰ</m:mi> <m:mi>s</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>μ</m:mi> <m:mi>ε</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {{\mathscr{E}}_{\varepsilon}^{s}(\mu):={\mathscr{E}}^{s}(\mu_{\varepsilon})} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>μ</m:mi> <m:mi>ε</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:hre