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On the axially symmetric solutions to the spatially homogeneous Landau equation

Jin Woo Jang, Junha Kim

TL;DR

The paper tackles the spatially homogeneous Landau equation in the grazing-collision regime, proving the existence of axisymmetric measure-valued weak solutions for hard potentials with axisymmetric initial data in $\mathcal{P}_p(\mathbb{R}^3)$, $p\ge 2$, and showing instantaneous analyticity for any $t>0$ unless the initial data is a Dirac mass. It introduces a regularization framework with a smooth, axisymmetric linearized operator, establishes uniform-in-$\varepsilon$ a priori estimates, and uses Schauder fixed-point arguments to construct axisymmetric solutions that preserve symmetry in the limit. A key lemma rules out time-interval line-concentration, leveraging axisymmetry to promote the support to nondegenerate configurations, which then enables ellipticity arguments and smoothing. Consequently, the authors obtain analyticity for $t>0$ in the hard-potential case, and show that for soft potentials (including Maxwell molecules) axisymmetric line-concentrated data cannot persist, highlighting a clear distinction between regimes. Altogether, the work advances the understanding of axisymmetric, measure-valued solutions to the Landau equation and provides a rigorous path from regularized approximations to global regularity results with potential implications for kinetic theory and related PDEs.

Abstract

In this paper, we consider the spatially homogeneous Landau equation, which is a variation of the Boltzmann equation in the grazing collision limit. For the Landau equation for hard potentials in the style of Desvillettes-Villani (Comm. Partial Differential Equations, 2000), we provide the proof of the existence of axisymmetric measure-valued solution for any axisymmetric $\mathcal{P}_p(\mathbb{R}^3)$ initial profile for any $p\ge 2$. Moreover, we prove that if the initial data is not a single Dirac mass, then the solution instantaneously becomes analytic for any time $t>0$ in the hard potential case. In the soft potential and the Maxwellian molecule cases, we show that there are no solutions whose support is contained in a fixed line even for any given line-concentrated data.

On the axially symmetric solutions to the spatially homogeneous Landau equation

TL;DR

The paper tackles the spatially homogeneous Landau equation in the grazing-collision regime, proving the existence of axisymmetric measure-valued weak solutions for hard potentials with axisymmetric initial data in , , and showing instantaneous analyticity for any unless the initial data is a Dirac mass. It introduces a regularization framework with a smooth, axisymmetric linearized operator, establishes uniform-in- a priori estimates, and uses Schauder fixed-point arguments to construct axisymmetric solutions that preserve symmetry in the limit. A key lemma rules out time-interval line-concentration, leveraging axisymmetry to promote the support to nondegenerate configurations, which then enables ellipticity arguments and smoothing. Consequently, the authors obtain analyticity for in the hard-potential case, and show that for soft potentials (including Maxwell molecules) axisymmetric line-concentrated data cannot persist, highlighting a clear distinction between regimes. Altogether, the work advances the understanding of axisymmetric, measure-valued solutions to the Landau equation and provides a rigorous path from regularized approximations to global regularity results with potential implications for kinetic theory and related PDEs.

Abstract

In this paper, we consider the spatially homogeneous Landau equation, which is a variation of the Boltzmann equation in the grazing collision limit. For the Landau equation for hard potentials in the style of Desvillettes-Villani (Comm. Partial Differential Equations, 2000), we provide the proof of the existence of axisymmetric measure-valued solution for any axisymmetric initial profile for any . Moreover, we prove that if the initial data is not a single Dirac mass, then the solution instantaneously becomes analytic for any time in the hard potential case. In the soft potential and the Maxwellian molecule cases, we show that there are no solutions whose support is contained in a fixed line even for any given line-concentrated data.
Paper Structure (23 sections, 10 theorems, 106 equations)

This paper contains 23 sections, 10 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Previous Results
  4. Conservation laws
  5. Further properties of the Landau collision operator for hard potentials
  6. Gain of moments
  7. Ellipticity of $\overline{a_{ij}(v)}$
  8. Our main result and relevant previous results
  9. Key Lemma for our result
  10. Proof of Theorem \ref{['main thm']}
  11. Regularized System
  12. Regularized initial profiles
  13. Linear approximating equation
  14. Non-negativity and upper-bound estimate
  15. Parabolic regularity and Schauder's fixed point theorem
  16. ...and 8 more sections

Key Result

Theorem 1.2

Let $f$ be any weak solution of the Landau equation eq:L with initial datum $f_{in}\in L^1_2(\mathbb{R}^3)$, satisfying the decay of energy Then

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2: Theorems 3 and 6 of MR1737547
  • Proposition 1.3: Proposition 4 of MR1737547
  • Lemma 1.4: Lemma 9 of MR1737547
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8: Theorem 1 of MR2557895
  • Theorem 1.9: Theorems 3 and 5 of FOURNIER20211961
  • Lemma 1.10: Lemma 16 of FOURNIER20211961
  • ...and 5 more