Conditions at infinity for boltzmann's equation.

We consider here realistic conditions at infinity for solutions of the Boltzmann's equation, such as a pure Maxwellian equilibrium at infinity possibly with suitable boundary conditions on an exterior domain, different Maxwellian equilibria at +∞ and -;∞ in a tube-like situation and more generally conditions at infinity obtained from a fixed solution. In order to adapt the recent global existence and compactness results due to R.J. DiPerna and the author, we have to obtain some local a priori estimates on the mass, kinetic energy and entropy. And this is precisely what we achieve here by two different and new methods. The first one consists in using the relative entropy of solutions with respect to a fixed, possibly local, Maxwellian. This method allows to treat general collision kernels with angular cut-off and some of the conditions at infinity mentioned above. The second method is based upon a L¹ estimate and an extension of the entropy identity which uses a truncated H-functional. This method requires a “uniform integrability” condition on the collision kernel but allows to consider the most general conditions at infinity.