Global classical solutions to the ionic Vlasov-Poisson-Boltzmann system in a 3D periodic box

TL;DR

This paper proves the existence and exponential decay of global classical solutions to the ionic Vlasov-Poisson-Boltzmann system in a 3D periodic domain, addressing mathematical challenges from nonlinearities and ion-electron interactions.

Contribution

It establishes the first global well-posedness result with exponential decay for the ionic Vlasov-Poisson-Boltzmann system in a periodic setting, using advanced energy and elliptic estimates.

Findings

Existence of unique global classical solutions under small perturbations.

Solutions exhibit exponential decay over time.

Development of new coercivity inequalities for the linearized collision operator.

Abstract

We investigate the global well-posedness of the ionic Vlasov-Poisson-Boltzmann system which models the evolution of dilute collisional ions. This system distinguishes the electronic Vlasov-Poisson-Boltzmann system via an additional exponential nonlinearity in the coupled Poisson-Poincar\'{e} equation, which introduces essential mathematical difficulties. In a three-dimensional periodic box, We establish the existence of a unique global-in-time classical solution with an exponential decay under small initial perturbations of a global Maxwellian that preserve mass, momentum and energy conservation laws. Our approach combines a nonlinear energy method with quantitative nonlinear elliptic estimates and new coercivity inequalities for the linearized collision operator $\mathcal{L}$ in ion dynamics.