The nonlinear Landau limit of weakly interacting particles
Let $\Phi \in C_c^\infty(\mathbb{R}^3)$ be nonzero, nonnegative, radial and nonincreasing in the radius. Put $\epsilon = N^{-1/3}$ and periodize $\Phi_\epsilon(x) = \sum_{n \in \mathbb{Z}^3} \Phi((x + n)/\epsilon)$ on the unit torus $\mathbb{T}^3$. Start $N$ particles independently, with uniformly distributed positions and a smooth, positive, rapidly decaying velocity probability density $f_0$, and let them evolve by $$\dot X_i = V_i, \qquad \dot V_i = -\sqrt{\epsilon} \sum_{j \ne i} \nabla \Phi_\epsilon(X_i - X_j).$$ Let $f_N(t)$ be the one-particle velocity marginal at time $t$. The expected limit is the spatially homogeneous Landau equation $$\partial_t f(v) = \nabla_v \cdot \int_{\mathbb{R}^3} a(v - w)\bigl[f(w)\nabla_v f(v) - f(v)\nabla_w f(w)\bigr]\,dw, \qquad f(0) = f_0,$$ $$a(z) = \frac{1}{2}\int_{\mathbb{R}}\int_{\mathbb{R}^3} \nabla\Phi(y) \otimes \nabla\Phi(y + sz)\,dy\,ds,$$ whose tensor has the form $c_\Phi |z|^{-1}(I - z\otimes z/|z|^2)$ for $z \ne 0$. Bobylev, Pulvirenti and Saffirio showed, in this scaling, that the first-order correction to free flow converges to the corresponding term of the Landau equation (2013). For the hierarchy truncated at pair correlations, Velázquez and Winter proved convergence to the Landau equation on times of order one with interactions at small relative velocity cut off (2018), and Winter without the cutoff (2021); both results assume initial data close to a Maxwellian and a specific potential, with $\widehat\phi(k) = (1 + |k|^2)^{-3/2}$, and hold on a short macroscopic time interval.
Problem. Does $f_N(t)$ converge to $f(t)$ as $N \to \infty$, weakly and uniformly on each interval $[0, T]$ on which a classical solution $f$ exists, that is, $$\sup_{0 \le t \le T} \Bigl| \int \psi\,df_N(t) - \int \psi(v) f(t,v)\,dv \Bigr| \to 0$$ for every $\psi \in C_b(\mathbb{R}^3)$? The full particle system is meant; its hierarchy may not be truncated.
Source: The derivation of the Landau equation (Landau, 1936) from particle dynamics in the weak-coupling limit, as formulated by Bobylev, Pulvirenti and Saffirio (2013) and Winter (2021), §1. Stated by AIM — Open Applied Problems for N particles on the unit torus.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Landau equation, particle systems, weak-coupling limits
References
@article{landau1936,
author = {Landau, L. D.},
title = {Die kinetische {G}leichung f{\"u}r den {F}all {C}oulombscher {W}echselwirkung},
journal = {Phys. Z. Sowjetunion},
volume = {10},
year = {1936},
pages = {154},
}
@article{bobylev2013,
author = {Bobylev, A. V. and Pulvirenti, M. and Saffirio, C.},
title = {From particle systems to the {L}andau equation: a consistency result},
journal = {Comm. Math. Phys.},
volume = {319},
number = {3},
year = {2013},
pages = {683--702},
doi = {10.1007/s00220-012-1633-6},
eprint = {1207.5987},
archivePrefix = {arXiv},
}
@article{winter2021,
author = {Winter, Raphael},
title = {Convergence to the {L}andau equation from the truncated {BBGKY} hierarchy in the weak-coupling limit},
journal = {J. Differential Equations},
volume = {283},
year = {2021},
pages = {1--36},
doi = {10.1016/j.jde.2021.02.041},
eprint = {1905.05021},
archivePrefix = {arXiv},
}
@article{velazquez2018,
author = {Vel{\'a}zquez, Juan J. L. and Winter, Raphael},
title = {From a non-{M}arkovian system to the {L}andau equation},
journal = {Comm. Math. Phys.},
volume = {361},
number = {1},
year = {2018},
pages = {239--287},
doi = {10.1007/s00220-018-3092-1},
eprint = {1707.07544},
archivePrefix = {arXiv},
}
@misc{aim-newton-landau-weak-coupling,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The nonlinear Landau limit of weakly interacting particles}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/150-newton-landau-weak-coupling.md},
note = {Entry 150 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-00882, "The nonlinear Landau limit of weakly interacting particles", clopen.solutions.