Existence, uniqueness and smoothing estimates for spatially homogeneous Landau-Coulomb equation in $H^{-\f12}$ space with polynomial tail
Ling-Bing He, Jie Ji, Yue Luo
TL;DR
This work analyzes the spatially homogeneous Landau-Coulomb equation with Coulomb potential, establishing global existence and uniqueness for data near the negative Sobolev space $H^{-1/2}_3$ under polynomial tails, i.e., within $H^{-1/2}_3 \cap L^1_7 \cap L log L$. The authors develop a robust collision-operator framework based on dyadic localization in phase and frequency to obtain sharp coercivity and upper bounds, along with precise commutator estimates. They prove smoothing effects in weighted Sobolev spaces, showing a $t$-dependent gain of regularity that is sharp with respect to a toy-model intuition, and they characterize the tail behavior by proving propagation of exponential tails and Gevrey regularity, including optimal Gevrey index depending on the tail parameter. A striking feature is that polynomial tails yield $C^\infty$ smoothing in velocity for any $t>0$, but this does not extend to $H^\infty$ smoothing, highlighting a nuanced regularity picture for the Landau operator. Overall, the paper advances the understanding of regularity, tail propagation, and long-time behavior for the Landau-Coulomb equation using localization and energy methods that separate smoothing from uniqueness.
Abstract
We demonstrate that the spatially homogeneous Landau-Coulomb equation exhibits global existence and uniqueness around the space $H^{-\f12}_3\cap L^1_{7}\cap L\log L$. Additionally, we furnish several quantitative assessments regarding the smoothing estimates in weighted Sobolev spaces. As a result, we confirm that the solution exhibits a \( C^\infty \) but not \( H^\infty \) smoothing effect in the velocity variable for any positive time, when the initial data possesses a polynomial tail.