arXiv subjects

Jong-in Kim

Publications and source records attributed to Jong-in Kim.

3 recordsLinked to original sources

arXiv2026

Asymptotic behavior of large-amplitude solutions to the Boltzmann equation with soft interactions in $L^p_v L^\infty_x$ spaces

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setting is that the standard arguments used in [Ko-Lee-Park,2022], [Li,2022] for the nonlinear loss term are no longer applicable when dealing with time integration involving the collision frequency.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.

math.AP↗

arXiv2025

Global solutions in $L^{p}_{v}L^{\infty}_{x}$ for the Boltzmann equation in bounded domains

The existence theory for solutions to the Boltzmann equation in bounded domains has primarily been developed within uniformly bounded function classes, such as $L^{\infty}_{x,v}$, as in [Duan-Huang-Wang-Yang,2017], [Duan-Wang,2019], [Guo,2010]. In this paper, we investigate solutions in relaxed function spaces $L^{p}_{v}L^\infty_{x}$ for the initial-boundary value problem of the Boltzmann equation in bounded domains. We consider the case of hard potential under diffuse reflection boundary conditions and assume cutoff model. For large initial data in a weighted $L^{p}_{v}L^\infty_{x}$ space with small relative entropy, we construct unique global-in-time mild solution that converge exponentially to the global Maxwellian. A pointwise estimate for the gain term, bounded in terms of $L^p_v$ and $L^2_v$ norms, is essential to prove our main results. Relative to [Gualdani-Mischler-Mouhot,2017], our work provides an alternative perspective on convergence to equilibrium in the presence of boundary conditions.

math.AP↗

arXiv2025

On the large amplitude solution of the Boltzmann equation with large external potential and boundary effects

The Boltzmann equation is a fundamental equation in kinetic theory that describes the motion of rarefied gases. In this study, we examine the Boltzmann equation within a $C^1$ bounded domain, subject to a large external potential $Φ(x)$ and diffuse reflection boundary conditions. Initially, we prove the asymptotic stability of small perturbations near the local Maxwellian $μ_E (x,v)$. Subsequently, we demonstrate the asymptotic stability of large amplitude solutions with initial data that is arbitrarily large in (weighted) $L^\infty$, but sufficiently small in the sense of relative entropy. Specifically, we extend the results for large amplitude solutions of the Boltzmann equation (with or without external potential) [10, 11, 12, 23] to scenarios involving significant external potentials [19, 28] under diffuse reflection boundary conditions.

math.AP↗