The Landau equation is a fundamental kinetic model for weakly collisional plasmas. In a recent breakthrough, Guillén and Silvestre [1] showed that, for the spatially homogeneous single-species Landau equation with very soft potentials -3 ≤ γ < -2, where γ = -3 corresponds to Coulomb interactions, Fisher information is monotone nonincreasing along the flow. Combined with local-posedness, this yields global well-posedness and opens a new approach to regularity for kinetic equations.
In this talk, I will discuss what remains of this mechanism for multi-species Landau systems, where particles may have different masses. In the multi-species setting, we prove that Fisher information is monotone precisely for γ≥−√8, and that this threshold is sharp: for every γ<−√8, there exist smooth initial data for which Fisher information increases. Since the physically relevant Coulomb interaction corresponds to γ=−3<−√8, Fisher information is therefore not a Lyapunov functional for the multi-species Landau-Coulomb equation. This failure is caused by the loss of the particle-exchange symmetry that is crucial in the single-species argument.
I will then explain how global well-posedness can nevertheless be recovered for a limiting ion-electron system obtained in the infinite ion-mass regime. Combining Fisher information, relative entropy, and the entropy dissipation along ion-electron collisions leads to a new Lyapunov functional whose monotonicity yields uniform bounds on the Fisher information. This is joint work with Raphaël Winter and Havva Yoldaş [2].
References:
[1] — GUILLEN, Nestor and SILVESTRE, Luis. The Landau equation does not blow up. Acta Mathematica, 2025, vol. 234, no 2, p. 315-375.
[2] — JUNNÉ, Jonathan, WINTER, Raphael, and YOLDAŞ, Havva. On the existence of solutions to the multi-species Landau equation. arXiv preprint arXiv:2512.17082, 2025.
