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Global regularity and decay estimates for the relativistic Landau equation

Christopher Henderson, Stanley Snelson, Andrei Tarfulea, Maja Tasković

TL;DR

The paper analyzes the spatially inhomogeneous, large-data relativistic Landau equation $\partial_t f + \frac{p}{\langle p\rangle}\cdot \nabla_x f = \mathcal{Q}_{\rm RL}(f,f)$ and proves that solutions become $C^{\infty}$ in $(t,x,p)$ under zeroth-order a priori bounds, while also establishing propagation of polynomial and exponential decay in momentum. The authors develop a relativistic Schauder theory by reducing to a nonrelativistic problem via the velocity variable and employing Lorentz boosts to localize estimates, then bootstrap regularity by transferring smoothness to the nonlocal collision coefficients $a^f,b^f,c^f$. These results extend smoothing and decay phenomena to the large-data, inhomogeneous regime, providing a rigorous foundation for potential global existence and qualitative analyses beyond near-equilibrium. The work introduces Lorentzian Hölder spaces and a robust framework to handle the non-translation-invariant collision kernel, yielding a suite of regularity and decay tools tailored to relativistic kinetic equations. Overall, the paper advances the mathematical understanding of relativistic collisional transport, offering new techniques with potential applicability to related relativistic Fokker-Planck-type systems and their Lorentz-invariant properties.

Abstract

We consider the relativistic Landau equation in the spatially inhomogeneous, far-from-equilibrium regime. We establish regularity estimates of all orders, implying that solutions remain smooth for as long as some zeroth-order conditional bounds hold. We also prove that polynomial and exponential decay in the momentum variable is propagated forward in time. As part of our proof, we establish a Schauder estimate for linear relativistic kinetic equations, that may be of independent interest.

Global regularity and decay estimates for the relativistic Landau equation

TL;DR

The paper analyzes the spatially inhomogeneous, large-data relativistic Landau equation and proves that solutions become in under zeroth-order a priori bounds, while also establishing propagation of polynomial and exponential decay in momentum. The authors develop a relativistic Schauder theory by reducing to a nonrelativistic problem via the velocity variable and employing Lorentz boosts to localize estimates, then bootstrap regularity by transferring smoothness to the nonlocal collision coefficients . These results extend smoothing and decay phenomena to the large-data, inhomogeneous regime, providing a rigorous foundation for potential global existence and qualitative analyses beyond near-equilibrium. The work introduces Lorentzian Hölder spaces and a robust framework to handle the non-translation-invariant collision kernel, yielding a suite of regularity and decay tools tailored to relativistic kinetic equations. Overall, the paper advances the mathematical understanding of relativistic collisional transport, offering new techniques with potential applicability to related relativistic Fokker-Planck-type systems and their Lorentz-invariant properties.

Abstract

We consider the relativistic Landau equation in the spatially inhomogeneous, far-from-equilibrium regime. We establish regularity estimates of all orders, implying that solutions remain smooth for as long as some zeroth-order conditional bounds hold. We also prove that polynomial and exponential decay in the momentum variable is propagated forward in time. As part of our proof, we establish a Schauder estimate for linear relativistic kinetic equations, that may be of independent interest.
Paper Structure (32 sections, 39 theorems, 442 equations)

This paper contains 32 sections, 39 theorems, 442 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Comparison with classical Landau equation
  4. Relativistic Schauder estimate
  5. Related work
  6. Difficulties and proof ideas
  7. Decay
  8. Regularity
  9. Open problems
  10. Filling of vacuum
  11. Weakening the conditional assumptions in \ref{['t:main']}
  12. Short-time existence
  13. Notation
  14. Outline of the paper
  15. Preliminaries and known results
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $f$ be a classical solution of the relativistic Landau equation e.main on $[0,T]\times \mathbb R^3_x\times\mathbb R^3_p$ for some $T>0$. Assume that there are $E_0, F_0 > 0$ such that for all $(t,x) \in [0,T]\times\mathbb R^3$. Further, suppose that there are $k > 0$ and $M_k > 0$ such that the initial data $f_{\rm in}: \mathbb R^3_x\times\mathbb R^3_p\to [0,\infty)$ satisfies Then $f$ satis

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 57 more