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Local regularity for the space-homogeneous Landau equation with very soft potentials

François Golse, Cyril Imbert, Sehyun Ji, Alexis F. Vasseur

Abstract

This paper deals with the space-homogenous Landau equation with very soft potentials, including the Coulomb case. This nonlinear equation is of parabolic type with diffusion matrix given by the convolution product of the solution with the matrix $a_{ij} (z)=|z|^γ(|z|^2 δ_{ij} - z_iz_j)$ for $γ\in [-3,-2)$. We derive local truncated entropy estimates and use them to establish two facts. Firstly, we prove that the set of singular points (in time and velocity) for the weak solutions constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has zero $\mathscr{P}^{m_\ast}$ parabolic Hausdorff measure with $m_\ast:= \frac72 |2+γ|$. Secondly, we prove that if such a weak solution is axisymmetric, then it is smooth away from the symmetry axis. In particular, radially symmetric weak solutions are smooth away from the origin.

Local regularity for the space-homogeneous Landau equation with very soft potentials

Abstract

This paper deals with the space-homogenous Landau equation with very soft potentials, including the Coulomb case. This nonlinear equation is of parabolic type with diffusion matrix given by the convolution product of the solution with the matrix for . We derive local truncated entropy estimates and use them to establish two facts. Firstly, we prove that the set of singular points (in time and velocity) for the weak solutions constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has zero parabolic Hausdorff measure with . Secondly, we prove that if such a weak solution is axisymmetric, then it is smooth away from the symmetry axis. In particular, radially symmetric weak solutions are smooth away from the origin.
Paper Structure (71 sections, 23 theorems, 361 equations, 1 figure)

This paper contains 71 sections, 23 theorems, 361 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Presentation of the problem and main results
  3. Previous results and related works
  4. General results on the Landau equation.
  5. The Landau-Coulomb equation.
  6. The "isotropic" Landau-Coulomb model
  7. The De Giorgi method and partial regularity.
  8. The De Giorgi method and kinetic equations.
  9. Main steps in the proofs of Theorems A and B
  10. Outline of this article
  11. Notation.
  12. Constants.
  13. Acknowledgements.
  14. Diffusion, dissipation and interpolation
  15. Derivatives of the matrix field a
  16. ...and 56 more sections

Key Result

Lemma 2.1

Pick $M_0>m_0>0$ and $E_0,H_0>0$, and let $f \colon \mathbb R^3 \to[0,+\infty)$ be a measurable function such that There exists $\bar{\delta} \in (0,1)$ and $c_0,R_0 >0$ depending only on $m_0,M_0,E_0,H_0$ such that for all $\delta \in (0,\bar{\delta}]$, we have where $I$ is the $3 \times 3$ identity matrix.

Figures (1)

  • Figure 1: The intersection of the orthogonal projection of the ball $B_{3\varepsilon}(v_0)$ on the affine plane passing through the origin with direction $(\mathbb R e_3)^\perp$. The angle $\theta$ is $\arcsin\left(\frac{3\varepsilon}{\rho_0}\right)$.

Theorems & Definitions (55)

  • Definition 1.1
  • Lemma 2.1: Diffusion matrix
  • proof
  • Proposition 2.2: Local entropy dissipation estimate
  • proof
  • Lemma 2.3: Interpolation
  • proof
  • Lemma 2.4: Control of short-range interactions
  • proof
  • Lemma 2.5: Nonlinearization
  • ...and 45 more