Nonlinear Landau damping near a Maxwellian on the whole space
On $\mathbb{R}^3_x \times \mathbb{R}^3_v$ consider the repulsive Vlasov–Poisson system with neutralizing background $$\partial_t f + v \cdot \nabla_x f + E \cdot \nabla_v f = 0, \qquad E = -\nabla_x \Phi, \qquad -\Delta_x \Phi = \int_{\mathbb{R}^3} (f - \mu)\,dv,$$ where $\mu(v) = (2\pi)^{-3/2} e^{-|v|^2/2}$ is the Maxwellian and $E$ is the Coulomb field, decaying at spatial infinity. For $\lambda > 0$ let $$\|h\|_{\mathcal{A}_\lambda} = \sum_{\alpha, \beta \in \mathbb{N}_0^3} \frac{\lambda^{|\alpha| + |\beta|}}{\alpha!\,\beta!} \left\| \langle x \rangle^4 \langle v \rangle^6 \partial_x^\alpha \partial_v^\beta h \right\|_{L^2_{x,v}}, \qquad \langle z \rangle = (1 + |z|^2)^{1/2}.$$ On the torus $\mathbb{T}^3_x$, Mouhot and Villani proved nonlinear Landau damping: small analytic perturbations of homogeneous equilibria satisfying the Penrose stability condition give global solutions whose electric field decays exponentially, and Gevrey perturbations close to analytic give decay at a stretched-exponential rate (2011). On $\mathbb{R}^3_x$ the Penrose condition fails at low spatial frequencies for every homogeneous equilibrium. For the system linearized around the Maxwellian, Bedrossian, Masmoudi and Mouhot showed that the electric field splits into a very weakly damped Klein–Gordon-type part for long waves and a Landau-damped part (2022). Ionescu, Pausader, Wang and Widmayer proved that small, smooth, localized perturbations of the Poisson equilibrium $\pi^{-2}(1 + |v|^2)^{-2}$, which has polynomial tails, lead to global solutions that scatter to free transport, with an electric field decaying in $L^\infty$ at a polynomial rate (2024).
Problem. Does every fixed $\lambda > 0$ admit an $\varepsilon_\lambda > 0$ such that all initial data $f_0 = \mu + h_0 \ge 0$ with $\int h_0 \, dx \, dv = 0$ and $\|h_0\|_{\mathcal{A}_\lambda} \le \varepsilon_\lambda$ generate a global classical solution satisfying $\|E(t)\|_{L^\infty_x} \to 0$ as $t \to \infty$? No exponential rate of decay in time is required.
Source: Stated by AIM — Open Applied Problems, in a localized analytic form, after the question of nonlinear stability of the Maxwellian and other homogeneous equilibria for the Vlasov–Poisson system on the whole space; see Ionescu, Pausader, Wang and Widmayer (2023, 2024) and Wang, Xiao and Xiong (2026).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Landau damping, Vlasov–Poisson equations
References
@article{ionescu2023,
author = {Ionescu, Alexandru D. and Pausader, Benoit and Wang, Xuecheng and Widmayer, Klaus},
title = {On the stability of homogeneous equilibria in the {V}lasov--{P}oisson system on $\mathbb{R}^3$},
journal = {Class. Quantum Grav.},
volume = {40},
number = {18},
year = {2023},
pages = {185007},
doi = {10.1088/1361-6382/acebb0},
eprint = {2305.11166},
archivePrefix = {arXiv},
}
@article{ionescu2024,
author = {Ionescu, Alexandru D. and Pausader, Benoit and Wang, Xuecheng and Widmayer, Klaus},
title = {Nonlinear {L}andau damping for the {V}lasov--{P}oisson system in $\mathbb{R}^3$: the {P}oisson equilibrium},
journal = {Ann. PDE},
volume = {10},
number = {1},
year = {2024},
pages = {Paper No. 2},
doi = {10.1007/s40818-023-00161-w},
eprint = {2205.04540},
archivePrefix = {arXiv},
}
@misc{wang2026,
author = {Wang, Yi and Xiao, Meixia and Xiong, Hang},
title = {Nonlinear {L}andau damping for the two-species screened {V}lasov--{P}oisson system with large initial distributions},
year = {2026},
eprint = {2603.16767},
archivePrefix = {arXiv},
}
@article{mouhot2011,
author = {Mouhot, Cl{\'e}ment and Villani, C{\'e}dric},
title = {On {L}andau damping},
journal = {Acta Math.},
volume = {207},
number = {1},
year = {2011},
pages = {29--201},
doi = {10.1007/s11511-011-0068-9},
eprint = {0904.2760},
archivePrefix = {arXiv},
}
@article{bedrossian2022,
author = {Bedrossian, Jacob and Masmoudi, Nader and Mouhot, Cl{\'e}ment},
title = {Linearized wave-damping structure of {V}lasov--{P}oisson in $\mathbb{R}^3$},
journal = {SIAM J. Math. Anal.},
volume = {54},
number = {4},
year = {2022},
pages = {4379--4406},
doi = {10.1137/20M1386141},
eprint = {2007.08580},
archivePrefix = {arXiv},
}
@misc{aim-maxwellian-whole-space-landau-damping,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Nonlinear Landau damping near a Maxwellian on the whole space}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/382-maxwellian-whole-space-landau-damping.md},
note = {Entry 382 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00615, "Nonlinear Landau damping near a Maxwellian on the whole space", clopen.solutions.