Steady equilibria of the Vlasov–Maxwell–Landau system on the three-torus

What are the smooth positive steady single-species Vlasov–Maxwell–Landau solutions on the three-torus with Coulomb collisions, Schwartz velocity decay and the stated polynomial score bound? The target classification is a spatially uniform Maxwellian at ion density, zero electric field and constant magnetic field.

Why it matters and what counts as progress

Why it matters: A classification identifies whether nonuniform equilibria survive in the coupled electromagnetic collision system.

What counts as progress: A proof exhausting every equilibrium under the stated decay, regularity and score assumptions, or a nonuniform solution meeting them.

How the problem developed

2011 · Partial result [1]
Yan Guo Established global solutions near Maxwellians for the periodic Vlasov–Poisson–Landau system, a related electrostatic model rather than the coupled Maxwell system.
16 Mar 2026 · Classification claim [2]
Vilin Ilin Reported that smooth positive steady solutions under specified decay and score bounds are spatially uniform Maxwellians with zero electric field and constant magnetic field.

AI contribution

Semi-Autonomous Formalization of the Vlasov-Maxwell-Landau Equilibrium

Vasily Ilin

For smooth positive steady single-species Vlasov–Maxwell–Landau solutions on the three-torus with Coulomb collisions, Schwartz velocity decay and a polynomial score bound, the stated Lean theorem yields a spatially uniform Maxwellian at ion density, zero electric field and constant magnetic field.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4 + human)

Human understandingFull