Mathematics Colloquium

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The Boltzmann and Landau equations are used to describe the evolution of the density of particles in a gas or plasma. It has been well-known for a long time that the entropy associated to the solution is monotone with respect to time. We discuss recent results for the space homogeneous case establishing the monotonicity in time also for the Fisher information, obtaining a second Lyapunov function for these equations. As a consequence, we deduce that the solutions to these equations remain smooth globally in time in all cases of interest that were previously not well understood, including the important case of the Landau equation with Coulomb potentials.