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Regularizing effect of the spatially homogeneous Landau equation with soft potential

Xiao-Dong Cao, Chao-Jiang Xu, Yan Xu

TL;DR

The paper analyzes the spatially homogeneous Landau equation with soft potentials near Maxwellian equilibrium by linearizing around μ and employing a weighted energy framework with an exponential velocity weight to overcome degeneracy. It proves that, for γ ∈ (−3,0) and small perturbations, the solution is analytic in time and exhibits Gelfand–Shilov smoothing in velocity, with the smoothing index σ = max{1, (b−γ)/(2b)} determined by the chosen weight parameter b ∈ (0,2]. The key contributions include new coercivity, trilinear, and commutator estimates for the Landau operators, a robust existence theory for small data, and a detailed induction argument that yields time analyticity and velocity GS regularization. These results extend regularity theory to soft potentials, providing sharp quantitative smoothing rates and reinforcing the near-equilibrium behavior of the Landau dynamics. The techniques have potential implications for long-time behavior and the precise regularity of solutions in kinetic theory.

Abstract

This paper investigates the Cauchy problem of the spatially homogeneous Landau equation with soft potential under the perturbation framework to global equilibrium. We prove that the solution to the Cauchy problem exhibits analyticity in the time variable and the Gelfand-Shilov regularizing effect in the velocity variables.

Regularizing effect of the spatially homogeneous Landau equation with soft potential

TL;DR

The paper analyzes the spatially homogeneous Landau equation with soft potentials near Maxwellian equilibrium by linearizing around μ and employing a weighted energy framework with an exponential velocity weight to overcome degeneracy. It proves that, for γ ∈ (−3,0) and small perturbations, the solution is analytic in time and exhibits Gelfand–Shilov smoothing in velocity, with the smoothing index σ = max{1, (b−γ)/(2b)} determined by the chosen weight parameter b ∈ (0,2]. The key contributions include new coercivity, trilinear, and commutator estimates for the Landau operators, a robust existence theory for small data, and a detailed induction argument that yields time analyticity and velocity GS regularization. These results extend regularity theory to soft potentials, providing sharp quantitative smoothing rates and reinforcing the near-equilibrium behavior of the Landau dynamics. The techniques have potential implications for long-time behavior and the precise regularity of solutions in kinetic theory.

Abstract

This paper investigates the Cauchy problem of the spatially homogeneous Landau equation with soft potential under the perturbation framework to global equilibrium. We prove that the solution to the Cauchy problem exhibits analyticity in the time variable and the Gelfand-Shilov regularizing effect in the velocity variables.

Paper Structure

This paper contains 6 sections, 12 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Schema of proof for the main theorem
  3. Analysis of Landau operators
  4. Existence and Energy estimate
  5. Analyticity in time variable
  6. Gelfand-Shilov Regularizing Effect

Key Result

Theorem 1.1

For the soft potential $-3<\gamma<0$, assume that there exist positive constants $c_{0}$ and $0<b\le 2$ such that for any $f_{0}\in L^{2}(\mathbb R^{3})$ satisfying $\|\omega_0 f_{0}\|_{L^{2}(\mathbb R^{3})}$ small enough, then the Cauchy problem 1-2 admits a unique solution $\omega_{t} f\in \mathca and

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 10 more