# Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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## The Vlasov–Poisson–Landau system in the weakly collisional regimeHTML articles powered by AMS MathViewer

by Sanchit Chaturvedi, Jonathan Luk and Toan T. Nguyen
J. Amer. Math. Soc. 36 (2023), 1103-1189
DOI: https://doi.org/10.1090/jams/1014
Published electronically: January 10, 2023

## Abstract:

Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a $3$-torus, i.e. \begin{align*} \partial _t F(t,x,v) + v_i \partial _{x_i} F(t,x,v) + E_i(t,x) \partial _{v_i} F(t,x,v) = \nu Q(F,F)(t,x,v),\\ E(t,x) = \nabla \Delta ^{-1} (\int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v - {{\int }\llap {-}}_{\mathbb T^3} \int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v \, \mathrm {d} x), \end{align*} with $\nu \ll 1$. We prove that for $\epsilon >0$ sufficiently small (but independent of $\nu$), initial data which are $O(\epsilon \nu ^{1/3})$-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as $t\to \infty$. The solutions exhibit uniform-in-$\nu$ Landau damping and enhanced dissipation.

Our main result is analogous to an earlier result of Bedrossian for the Vlasov–Poisson–Fokker–Planck equation with the same threshold. However, unlike in the Fokker–Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo’s weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation.

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Bibliographic Information
• Sanchit Chaturvedi
• Affiliation: Department of Mathematics, Stanford University, 450 Jane Stanford Way, Bldg. 380, Stanford, California 94305
• MR Author ID: 1440099
• Email: sanchat@stanford.edu
• Jonathan Luk
• Affiliation: Department of Mathematics, Stanford University, 450 Jane Stanford Way, Bldg. 380, Stanford, California 94305
• MR Author ID: 916843
• Email: jluk@stanford.edu
• Toan T. Nguyen
• Affiliation: Department of Mathematics, Penn State University, State College, Pennsylvania 16802
• MR Author ID: 718370
• Email: nguyen@math.psu.edu
• Received by editor(s): April 26, 2021
• Received by editor(s) in revised form: July 4, 2022
• Published electronically: January 10, 2023
• Additional Notes: The first and second authors were supported by the NSF grant DMS-2005435. The second author was also supported by a Terman Fellowship. The third author was partly supported by the NSF under grant DMS-1764119, an AMS Centennial fellowship, and a Simons fellowship.