Global mild solutions of the non-cutoff Vlasov–Poisson–Boltzmann system

This paper is concerned with the Cauchy problem on the Vlasov–Poisson–Boltzmann system in the torus domain. The Boltzmann collision kernel is assumed to be angular non-cutoff with $0 \leq \gamma \lt 1$ and $1/2 \leq s \lt 1$, where $\gamma, s$ are two parameters describing the kinetic and angular singularities, respectively. We obtain the global-in-time unique mild solutions, and prove that the solutions converge to the global Maxwellian with the large-time decay rate of $\mathcal{O}(e^{-\lambda t})$ in the $L^1_k L^2_v$-norm for some $\lambda \gt 0$. Furthermore, we justify the property of propagation of regularity of solutions in the spatial variable.

Keywords

Vlasov–Poisson–Boltzmann system, non-cutoff, mild regularity, global existence, time decay

2010 Mathematics Subject Classification

35B35, 35Q20, 35Q83

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Received 13 February 2023

Accepted 8 May 2023

Published 7 December 2023