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<p>In this paper we introduce the notion of parabolic <i>α</i>-Riesz flow, for <i>α</i> ∈ (0, <i>d</i>), extending the notion of <i>s</i>-fractional heat flows to negative values of the parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(s=-{\alpha \over 2}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mrow> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. Then, we determine the limit behaviour of these gradient flows as <i>α</i> → 0<sup>+</sup> and <i>α</i> → <i>d</i><sup>−</sup>.</p><p>To this end we provide a preliminary Γ-convergence expansion for the Riesz interaction energy functionals. Then we apply abstract stability results for uniformly <i>λ</i>-convex functionals which guarantee that Γ-convergence commutes with the gradient flow structure.</p>

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Parabolic α-Riesz flows and limit cases α → 0+, α → d−

  • 期刊论文

  • 发表日期：

    2025年11月12日

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  • Lucia De Luca,
  • Massimiliano Morini,
  • Marcello Ponsiglione,
  • Emanuele Spadaro

摘要

In this paper we introduce the notion of parabolic α-Riesz flow, for α ∈ (0, d), extending the notion of s-fractional heat flows to negative values of the parameter \(s=-{\alpha \over 2}\) s = − α 2 . Then, we determine the limit behaviour of these gradient flows as α → 0+ and α → d−.

To this end we provide a preliminary Γ-convergence expansion for the Riesz interaction energy functionals. Then we apply abstract stability results for uniformly λ-convex functionals which guarantee that Γ-convergence commutes with the gradient flow structure.