STRICT ENTROPY PRODUCTION BOUNDS AND STABILITY OF THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR THE BOLTZMANN-EQUATION

We first consider the Boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates. This is the simplest Boltzmann equation having the compressible Euler equations as a scaling limit. For it we prove a stability result for the H-theorem which says that when the entropy production is small, the solution of the spatially homogeneous Boltzmann equation is necessarily close to equilibrium in the entropic sense, and therefore strong L1 sense. We use this to prove that solutions to the spatially homogeneous Boltzmann equation converge to equilibrium in the entropic sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes. Every initial condition with finite entropy and p th velocity moment for some p > 2 belongs to such a class. We then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below, which covers a wide class of slightly modified physical collision kernels. These results are the basis of a study of the relation between scaling limits of solutions of the Boltzmann equation and hydrodynamics which will be developed in subsequent papers; the program is described here.