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

Abstract: In this paper, we study the global well-posedness of the Boltzmann equation within the $L_{v}<sup>{p}L_{x}<sup>{\infty}$ framework for soft potential models with angular cutoff in a periodic box $\mathbb{T}<sup>3$. 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 $L_v^p L_x<sup>\infty$ setting is that the standard arguments used in [Ko-Lee-Park,2022], [Li,2022] for the nonlinear loss term are no longer applicable when dealing with time integration involving the collision frequency.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 $L_v<sup>p$ and $L_v<sup>\ell$ (for some $\ell &lt;p$) 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.