The massless Boltzmann equation in Minkowski spacetime

Technical Summary: The Massless Boltzmann Equation in Minkowski Spacetime

Problem Statement
This paper investigates the existence of solutions to the spatially homogeneous, massless relativistic Boltzmann equation in a fixed Minkowski spacetime background. The study focuses on a specific range of collision cross-sections defined by $\sigma = \varrho^\gamma$, where $\varrho$ is the relative momentum of colliding particles and $-2 < \gamma \leq 1$. This range encompasses both hard interactions ($\gamma \geq 0$) and soft interactions ($\gamma < 0$).

The authors highlight that while the massive relativistic Boltzmann equation has been extensively studied regarding local and global existence, the massless case presents unique mathematical challenges. Specifically, the energy of massless particles is not bounded away from zero, leading to a singularity in the collision term at momentum $p=0$. Furthermore, the distribution function is supported on the light cone rather than a smooth mass hyperboloid. Existing literature on the massless case is sparse, with previous results often restricted to specific cosmological backgrounds, bounded cross-sections, or specific ranges of $\gamma$.

Methodology
The authors employ a standard cut-off approach to handle the unbounded nature of the collision cross-sections $\sigma = \varrho^\gamma$. They introduce modified Boltzmann equations where the collision kernel is truncated:

• Hard Interactions ($0 \leq \gamma \leq 1$): The cross-section is cut off for $\varrho \leq n$.
• Soft Interactions ($-2 < \gamma < 0$): The cross-section is cut off for $\varrho \geq n^{-1}$.

The analysis proceeds in two main stages:

1. A Priori Estimates for Cut-off Equations: The authors establish uniform bounds for the solutions of the modified equations ($f_n$) with respect to the cut-off parameter $n$.

   • For hard interactions, they derive a Povzner-type inequality (Lemma 2) tailored to massless particles. This inequality controls the growth of weighted $L^1$ norms ($\|f\|_{L^1_r}$) and is crucial for proving global existence.
   • For soft interactions, where a Povzner-type inequality is unavailable, they utilize singular weights to control the singularity at $p=0$. They employ weighted $L^\infty$ norms ($\|f\|_{L^\infty_w}$ with $w(p) = p_0 e^{p_0}$) and weighted $L^1$ norms with negative powers ($L^1_{-2}$). Lemma 3 provides estimates for integrals involving post-collision momenta to handle the singularities arising from the massless condition.

2. Convergence to the Original Equation: Using the uniform bounds derived for the cut-off solutions, the authors prove that the sequence $f_n$ converges in appropriate function spaces ($L^1 \cap L^1_r$ or $L^1 \cap L^1_{-1}$) as $n \to \infty$. This convergence yields a solution to the original, uncut Boltzmann equation.

Key Contributions and Results

• Global Existence for Hard Interactions (Theorem 1):
  For initial data $f_0 \in L^1 \cap L^1_2$ with $0 \leq \gamma \leq 1$, the authors prove the existence of a unique, non-negative global solution $f \in C^1([0, \infty); L^1 \cap L^1_1)$. Furthermore, if the initial data possesses higher moments ($f_0 \in L^1_r$ for $r \geq 1$), these moments remain bounded for any finite time interval. This result relies on the newly derived Povzner-type inequality for massless particles.

• Local Existence for Soft Interactions (Theorem 2):
  For initial data $f_0 \in L^1 \cap L^1_{-2} \cap L^\infty_w$ with $-2 < \gamma < 0$, the authors establish the existence of a unique, non-negative local solution $f \in C^1([0, T]; L^1 \cap L^1_{-1})$. The solution satisfies uniform bounds on the weighted norms over the time interval $[0, T]$. The proof requires the use of singular weights to manage the $p=0$ singularity inherent in soft interactions.

• Extension of Collision Cross-Section Ranges:
  The paper extends the known existence results for the massless Boltzmann equation to a wider range of $\gamma$ than previously available in fixed Minkowski backgrounds, specifically covering the full range $-2 < \gamma \leq 1$ (with global existence for $\gamma \geq 0$ and local for $\gamma < 0$).

Significance and Claims
The authors position this work as filling a significant gap in the literature regarding the mathematical theory of the massless Boltzmann equation. They note that existing proofs for the massive case cannot be directly transferred to the massless case due to the lack of a lower bound on energy and the geometry of the light cone.

The paper explicitly distinguishes its results from studies on the stability of Minkowski spacetime coupled to the Einstein equations; here, the background is fixed. The motivation for studying this system stems from cosmological settings where massive particles near the initial singularity behave effectively as massless particles (with a pressure-to-energy density ratio approaching $1/3$). The authors note that their previous work on the massless Einstein–Boltzmann system in cosmological settings relied on monotonicity arguments involving the scale factor, which are not applicable in the static Minkowski case ($R=1$). Consequently, the development of new analytical tools, such as the specific Povzner-type inequality for hard interactions and the singular weight techniques for soft interactions, represents a necessary "new start" for existence proofs in this regime.

The results are presented as rigorous existence proofs for a fundamental equation in relativistic kinetic theory, providing a foundation for further study of massless particle dynamics in special relativistic settings.