Equilibria of the Vlasov–Maxwell–Landau system

The problem asks whether the stated assumptions force every steady Coulomb Vlasov–Maxwell–Landau state to be a spatially uniform Maxwellian, with $E=0$ and constant $B$.

Known results

• For the Landau equation alone, entropy dissipation and a suitable Korn inequality yield global Maxwellians, including on the torus; this does not classify the coupled electromagnetic system.

2026 formalization claim

A paper claims a complete Lean formalization of the exact equilibrium characterization on $\mathbb{T}^{3}\times\mathbb{R}^{3}$, with no remaining sorrys. It reports that a definition mismatch concerning smoothness was found and corrected to finite $C^{3}$ velocity and $C^{2}$ spatial assumptions; the authors say the theorem still holds. Gemini DeepThink generated the initial proof, while other tools formalized auxiliary steps. The claim remains unverified independently.

Current status (as of September 2026): A corrected Lean formalization claims the characterization, but independent verification is absent, so the problem remains unsettled.