On the monotonicity of the entropy production in the Landau-Maxwell equation

We study the homogeneous Landau equation with Maxwell molecules and prove that the entropy production is non-increasing provided the directional temperatures are well-distributed and the solution admits a moment of order $\mathrm{\ell }$, for some $\mathrm{\ell }$ arbitrarily close to $2$. It implies that for an initial condition with finite moment of order $\mathrm{\ell }$, the entropy production is guaranteed to be non-increasing after a certain time, that we explicitly compute. This is the first partial answer to a conjecture made by Henry P. McKean in 1966 on the sign of the time-derivatives of the entropy. Without moment assumptions, we obtain a possibly sharp short-time regularization rate for the entropy production, and exponential decay for large times.