All-orders sign alternation for entropy along the Gaussian heat flow

Determine whether all successive time derivatives of the entropy along the Gaussian heat flow have alternating signs, thereby resolving the full all-orders version of McKean’s conjecture for that flow.

Background

McKean’s conjecture concerns alternating-sign properties of successive time derivatives of entropy along dissipative evolutions. For the Gaussian heat flow, the cited work of Carlen and Gangbo establishes the expected sign pattern through the third and fourth derivatives but explicitly does not settle the corresponding statement at every order.

The unresolved issue is whether the alternating-sign inequality continues to hold for every derivative order in the Gaussian heat-flow setting. This question is distinct from the paper’s principal result, which disproves entropy-dissipation monotonicity for the spatially homogeneous Landau–Coulomb equation and, through the grazing-collision limit, for associated Boltzmann equations.

References

In, the authors prove that the third and fourth derivatives of entropy along the Gaussian heat flow have alternating signs, thereby confirming McKean’s conjecture up to fourth order, while leaving the full all-orders conjecture open.

— A counterexample to McKean's conjecture for the Landau-Coulomb equation  (2609.03847 - Junné et al., 3 Sep 2026) in Introduction, paragraph titled “State of the art”