Asymptotic behavior of large-amplitude solutions to the Boltzmann equation with soft interactions in $L^p_v L^\infty_x$ spaces
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| Format: | Preprint |
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2026
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| author | Kim, Jong-in Ko, Gyounghun |
| author_facet | Kim, Jong-in Ko, Gyounghun |
| contents | In this paper, we study the global well-posedness of the Boltzmann equation within the $L_{v}^{p}L_{x}^{\infty}$ framework for soft potential models with angular cutoff in a periodic box $\mathbb{T}^3$. By using a time-involved weight function, inspired by the works of [Liu-Yang,2017], [Duan-Yang-Zhao,2013], [Ko-Lee-Park,2022], we overcome the absence of a spectral gap. An analytical difficulty in the $L_v^p L_x^\infty$ setting is that the standard arguments used in [Ko-Lee-Park,2022], [Li,2022] for the nonlinear loss term are no longer applicable when dealing with time integration involving the collision frequency. To resolve this, we introduce a modified solution operator. Furthermore, we control the nonlinear gain term by deriving pointwise estimates bounded by $L_v^p$ and $L_v^\ell$ (for some $\ell <p$) norms. Thanks to the smallness of the initial relative entropy and Grönwall's inequality, we prove the global existence of unique solutions for large-amplitude initial data and obtain a sub-exponential convergence rate toward equilibrium. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11903 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of large-amplitude solutions to the Boltzmann equation with soft interactions in $L^p_v L^\infty_x$ spaces Kim, Jong-in Ko, Gyounghun Analysis of PDEs 35Q20, 82C40 In this paper, we study the global well-posedness of the Boltzmann equation within the $L_{v}^{p}L_{x}^{\infty}$ framework for soft potential models with angular cutoff in a periodic box $\mathbb{T}^3$. By using a time-involved weight function, inspired by the works of [Liu-Yang,2017], [Duan-Yang-Zhao,2013], [Ko-Lee-Park,2022], we overcome the absence of a spectral gap. An analytical difficulty in the $L_v^p L_x^\infty$ setting is that the standard arguments used in [Ko-Lee-Park,2022], [Li,2022] for the nonlinear loss term are no longer applicable when dealing with time integration involving the collision frequency. To resolve this, we introduce a modified solution operator. Furthermore, we control the nonlinear gain term by deriving pointwise estimates bounded by $L_v^p$ and $L_v^\ell$ (for some $\ell <p$) norms. Thanks to the smallness of the initial relative entropy and Grönwall's inequality, we prove the global existence of unique solutions for large-amplitude initial data and obtain a sub-exponential convergence rate toward equilibrium. |
| title | Asymptotic behavior of large-amplitude solutions to the Boltzmann equation with soft interactions in $L^p_v L^\infty_x$ spaces |
| topic | Analysis of PDEs 35Q20, 82C40 |
| url | https://arxiv.org/abs/2603.11903 |