Rigorous results for conservation equations and trend to equilibrium in space-inhomogeneous kinetic theory

Abstract

The well-posedness of the initial value problem for the Boltzmann equation means that we prove that there is a unique nonnegative solution preserving the energy and satisfying the entropy inequality, from a positive initial datum with finite energy and entropy. However, for general initial data, it is difficult, and until now not known, whether such a well-behaved solution can be constructed globally in time. The difficulty in doing this is obviously related to the nonlinearity of the collision operator and the apparent lack of conservation laws or a priori estimates preventing the solution from becoming singular in finite time.