Half-space problem on the Boltzmann equation with zero Mach number at infinity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911679263014912 |
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| author | Chen, Hongxu Chen, Jun-ling Duan, Renjun |
| author_facet | Chen, Hongxu Chen, Jun-ling Duan, Renjun |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_12914 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Half-space problem on the Boltzmann equation with zero Mach number at infinity Chen, Hongxu Chen, Jun-ling Duan, Renjun Analysis of PDEs 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. |
| title | Half-space problem on the Boltzmann equation with zero Mach number at infinity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.12914 |