Monotone Fisher information for Landau–Fermi–Dirac dynamics

Determine whether the Landau–Fermi–Dirac equation with Coulomb interactions admits a monotone Fisher information functional analogous to that of the classical Landau equation.

Background

The classical Landau equation has a Fisher-information structure that supports regularity and long-time analysis. The paper states that it is unresolved whether an analogous monotone functional exists for the Coulomb Landau–Fermi–Dirac equation, whose fermionic modification changes both the entropy structure and the steady states. Establishing such a functional could provide a useful tool for studying regularity and convergence to equilibrium.

References

In particular, it remains largely open whether regularity estimates can be made uniform in $$ in the Coulomb case, and whether LFD admits a monotone Fisher information functional analogous to that of the classical equation.

— Global smooth solutions to the inhomogeneous Landau-Fermi-Dirac equation  (2608.31071 - Golding et al., 31 Aug 2026) in Section 1, subsection “Previous Work on \eqref{e.LFD}”