Half-space problem on the Boltzmann equation with zero Mach number at infinity

TL;DR

This paper analyzes the long-time behavior and regularity of solutions to the Boltzmann equation in a half-space with zero Mach number at infinity, establishing decay rates and Gevrey regularity propagation.

Contribution

It constructs global solutions near Maxwellians, proves polynomial decay rates, and demonstrates propagation of Gevrey regularity in boundary value problems for the Boltzmann equation.

Findings

Established polynomial decay rates matching 2D heat equation behavior.

Proved propagation of Gevrey regularity in spatial and velocity variables.

Developed Fourier-space and macro-micro decomposition techniques for boundary problems.

Abstract

We study the long-time dynamics of the time-evolutionary Boltzmann equation with hard sphere collisions in the three-dimensional half-space \( \mathbb{R}^2 \times \mathbb{R}^+\), subject to diffuse reflection boundary conditions and small perturbations around a global Maxwellian equilibrium. The far-field velocity is assumed to be at rest; namely, we take the zero Mach number at infinity. In the first goal, we construct global-in-time low-regularity solutions near Maxwellians. We leverage time-decay properties along the two-dimensional tangential direction to establish polynomial decay rates of solutions matching the 2D heat equation. In the second goal, we further prove the propagation of Gevrey regularity: analyticity (Gevrey index 1) in the tangential spatial variable \(x_\parallel\), and Gevrey class with index 2 in the tangential velocity variable \(v_\parallel\), under suitably regular initial data. The proofs combine an \(L^1_k \cap L^p_k\) Fourier-space approach for decay estimates, macro-micro decomposition with \(L^2 - L^\infty\) frameworks adapted to unbounded domains, and weighted Gevrey norms to control regularity propagation, overcoming challenges from boundary effects and nonlinear interactions.