On Global-in-time Chaotic Weak Solutions of the Liouville Equation for Hard Spheres

Abstract

We outline a new method of construction of global-in-time weak solutions of the Liouville equation -and also of the associated BBGKY hierarchy -corresponding to the hard sphere singular Hamiltonian. Our method makes use only of geometric reflection arguments on phase space. As a consequence of our method, in the case of N = 2 hard spheres, we show for any chaotic initial data, the unique global-in-time weak solution F of the Liouville equation is realised aswhere R : S → S is a 'reflection-type operator' on Schwartz space, and f is a global-intime classical solution of the 1-particle free transport Liouville equation on T R 3 .