A counterexample to McKean's conjecture for the Landau-Coulomb equation

Abstract: We disprove McKean's conjecture, which asserts that the entropy dissipation is monotone nonincreasing along solutions, or equivalently that the entropy is convex in time, for the spatially homogeneous Landau-Coulomb equation. We provide an explicit counterexample which consists of a Maxwellian equilibrium under radially symmetric, smooth perturbation on an annulus of scale $R\gg 1$. Developing the derivative of the entropy dissipation in powers of $R$ reveals that the leading order terms can be positive for a suitably chosen perturbation. Remarkably, the counterexamples are close to equilibrium in relative entropy and standard Sobolev norms. Since the Landau equation arises from the Boltzmann equation in the grazing collisions limit, our counterexample also shows that McKean's conjecture is not true in general for the Boltzmann equation.