Global well-posedness of Vlasov-Poisson-Boltzmann equations with neutral initial data and small relative entropy
Zaihong Jiang, Yong Wang, Hang Xiong
TL;DR
The paper addresses global well-posedness for the two-species VPB system on a periodic torus, using a perturbation framework around the Maxwellian with a time-velocity weight to handle large amplitude data. It combines local existence with a robust set of a priori estimates in $L^{\\infty}$, $L^{\\infty}_xL^1_v$, and $W^{1,\\infty}$, along with hydrodynamic-microscopic decomposition and logarithmic field control, to prove global existence and exponential decay. The main contributions include global mild solutions for large-amplitude, nearly neutral data with small relative entropy, and exponential decay of fluctuations in $L^{\\infty}$ and $W^{1,\\infty}$. The results extend VPB well-posedness beyond small perturbations, providing quantitative decay rates and a framework that handles strong electric-field nonlinearities in a periodic setting, with potential implications for plasma dynamics and kinetic theory.
Abstract
The dynamics of dilute plasma particles such as electrons and ions can be modeled by the fundamental two species Vlasov-Poisson-Boltzmann equations, which describes mutual interactions of plasma particles through collisions in the self-induced electric field. In this paper, we are concerned with global well-posedness of mild solutions to these equations. We establish the global existence and uniqueness of mild solutions to the two species Vlasov-Poisson-Boltzmann equations on the torus for a class of initial data with bounded time-velocity-weighted $L^{\infty}$ norm under a nearly neutral condition, along with smallness conditions on the $L^1_xL^\infty_v$ norm and defects in mass, energy and entropy. These conditions allow the initial data to exhibit large amplitude oscillations. Due to the nonlinear effect of electric field, we consider the problem in $W^{1, \infty}_{x,v}$ with large amplitude data, new difficulty arises when establishing globally uniform $W^{1, \infty}_{x,v}$ bound, which has been overcome based on nearly neutral condition, time-velocity weight function and a logarithmic estimate. Moreover,the long-time behavior of solutions in $W^{1, \infty}_{x,v}$ norm, with exponential decay rates of convergence, is also obtained.