Existence, uniqueness and smoothing estimates for spatially homogeneous Landau-Coulomb equation in $H^{-\f12}$ space with polynomial tail

TL;DR

This paper proves global existence, uniqueness, and smoothing effects for the spatially homogeneous Landau-Coulomb equation in a specific Sobolev space, highlighting the solution's regularity and tail behavior.

Contribution

It establishes the first rigorous results on existence, uniqueness, and smoothing estimates for solutions with polynomial tails in the Landau-Coulomb equation.

Findings

Global existence and uniqueness in weighted Sobolev spaces.

Solutions become infinitely smooth in velocity for positive times.

Smoothing occurs without full H-infinity regularity.

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.