GENERALIZED VARIATIONAL-PRINCIPLES, GLOBAL WEAK SOLUTIONS AND BEHAVIOR WITH RANDOM INITIAL DATA FOR SYSTEMS OF CONSERVATION-LAWS ARISING INADHESION PARTICLE DYNAMICS

We study systems of conservation laws arising in two models of adhesio n particle dynamics. The first is the system of free particles which s tick under collision. The second is a system of gravitationally intera cting particles which also stick under collision. In both cases, mass and momentum are conserved at the collisions, so the dynamics is descr ibed by 2 x 2 systems of conservations laws. We show that for these sy stems, global weak solutions can be constructed explicitly using the i nitial data by a procedure analogous to the Lax-Oleinik variational pr inciple for scalar conservation laws. However, this weak solution is n ot unique among weak solutions satisfying the standard entropy conditi on. We also study a modified gravitational model in which, instead of momentum, some other weighted velocity is conserved at collisions. For this model, we prove both existence and uniqueness of global weak sol utions. We then study the qualitative behavior of the solutions with r andom initial data. We show that for continuous but nowhere differenti able random initial velocities, all masses immediately concentrate on points even though they were continuously distributed initially, and t he set of shock locations is dense.