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

Abstract

We study the homogeneous Landau equation with Maxwell molecules and prove that the entropy production is non-increasing when the directional temperatures are well-distributed. It implies that for any initial condition with finite mass and energy, the entropy production is guaranteed to be non-increasing after a certain explicit time. We also obtain exponential decay of the entropy production. We provide an explicit formula for the derivative of the entropy production which may be a starting point for future progress towards full monotonicity.

This manuscript has been majorly revised to fix a mistake in the formula of the previous version. This new proof is shorter and the result has been slightly improved