An example of non-uniqueness for solutions to the homogeneous Boltzmann equation

The paper deals with the spatially homogeneous Boltzmann equation for hard potentials. An example is given which shows that, even though it is known that there is only one solution that conserves energy, there may be other solutions for which the energy is increasing; uniqueness holds if and only if energy is assumed to be conserved 1. Introduction This paper deals with the spatially homogeneous Boltzmann equation 8 ? ! ? : @ @t f(t; v) = Q(f; f)(t; v) ; f(0; v) = f 0 (v) ; (1) where f(t; v) for each t gives the distribution of velocities of a spatially homogeneous gas. The operator Q(f; f) in the right hand side is the collision term, which here is assumed to describe hard sphere interaction. The details are given below. It is known that if Z R 3 f 0 (v)(1 + jvj 2 ) dv ! 1 ; then there is a unique energy-conserving solution: E(f(t; \Delta)) j Z R 3 f(t; v)jvj 2 dv = E(f 0 ) = const; for all t ? 0. This solution also conserves mass and momentum: Z R 3 f(t; v) dv = const;..