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<p>We study the derivation of the spatially homogeneous Landau equation from the mean-field limit of a conservative <i>N</i>-particle system, obtained by passing to the grazing limit on Kac’s walk in his program for the Boltzmann equation. Our result covers the full range of interaction potentials, including the physically important Coulomb case. This provides the first resolution of the propagation of chaos for a many-particle system approximating the Landau equation with Coulomb interactions, and the first extension of Kac’s program to the Landau equation in the soft potential regime. The convergence is established in weak, Wasserstein, and entropic senses, together with strong <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> convergence. To handle the singularity of soft potentials, we extend the duality approach of Bresch–Duerinckx–Jabin [<CitationRef CitationID="CR5">5</CitationRef>] and establish key functional inequalities, including an extended commutator estimate and a new second-order Fisher information estimate.</p>

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Kac’s Program for the Landau Equation

  • 期刊论文

  • 发表日期：

    2026年7月30日

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  • Xuanrui Feng,
  • Zhenfu Wang

摘要

We study the derivation of the spatially homogeneous Landau equation from the mean-field limit of a conservative N-particle system, obtained by passing to the grazing limit on Kac’s walk in his program for the Boltzmann equation. Our result covers the full range of interaction potentials, including the physically important Coulomb case. This provides the first resolution of the propagation of chaos for a many-particle system approximating the Landau equation with Coulomb interactions, and the first extension of Kac’s program to the Landau equation in the soft potential regime. The convergence is established in weak, Wasserstein, and entropic senses, together with strong \(L^1\) L 1 convergence. To handle the singularity of soft potentials, we extend the duality approach of Bresch–Duerinckx–Jabin [5] and establish key functional inequalities, including an extended commutator estimate and a new second-order Fisher information estimate.