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

TL;DR

This paper proves that under certain conditions, the entropy production in the Landau-Maxwell equation is non-increasing, providing partial answers to a longstanding conjecture and establishing decay rates.

Contribution

It establishes conditions under which entropy production is non-increasing and provides explicit time bounds, addressing a conjecture from 1966.

Findings

Entropy production is non-increasing with well-distributed directional temperatures.

Explicit time after which entropy production becomes non-increasing.

Exponential decay of entropy production for large times.

Abstract

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 $\ell$, for some $\ell$ arbitrarily close to $2$. It implies that for an initial condition with finite moment of order $\ell$, 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.