Curvature-Enhanced Dynamics and Exponential Decay of the Non-Cutoff Boltzmann Equation on Riemannian Manifolds

TL;DR

This paper studies the exponential decay of solutions to the non-cutoff Boltzmann equation on compact Riemannian manifolds, revealing how geometry and collision kernel singularities influence long-term behavior.

Contribution

It introduces new decay estimates for hydrodynamic quantities on curved spaces, extending kinetic theory analysis to Riemannian manifolds with singular collision kernels.

Findings

Density, momentum, and energy decay exponentially over time.

Decay rates depend on manifold curvature and collision kernel singularity.

Conditions for exponential decay with angular singularities are established.

Abstract

In this work, we investigate the long-time behavior of solutions to the non-cutoff Boltzmann equation on compact Riemannian manifolds with bounded Ricci curvature. The paper introduces new results on the exponential decay of hydrodynamic quantities, such as density, momentum, and energy fields, influenced by both the curvature of the manifold and singularities in the collision kernel. We demonstrate that for initial data in $H^s_x \times L^p_v$, the solutions exhibit sharp exponential decay rates in Sobolev norms, with the decay rate determined by the manifold's geometry and the regularity of the kernel. Specifically, we prove that the density $\rho(t, x)$, momentum $\mathbf{m}(t, x)$, and energy field $E(t, x)$ all decay exponentially in time, with decay rates that depend on the manifold's curvature and the nature of the collision kernel's singularity. Additionally, we address the case of angular singularities in the collision kernel, providing conditions under which the exponential decay persists. The analysis combines energy methods, Fourier analysis, and coercivity estimates for the collision operator, extended to curved geometries. These results extend the understanding of dissipation mechanisms in kinetic theory, especially in curved settings, and offer valuable insights into the behavior of rarefied gases and plasma flows in non-Euclidean environments. The findings have applications in plasma physics, astrophysics, and the study of rarefied gases, opening new directions for future research in kinetic theory and geometric analysis.