Dissipative solutions to Hamiltonian systems

We extend the notion of dissipative particle solutions [5] to the case of Hamiltonian flow in the space of probability measures mu is an element of P(Rd x Rd) in the sense of [3]. The Hamiltonian is of the formH(mu) = V (q, p)mu(dqdp) +1W(q, p, q ', p ')mu(dqdp)mu(dq ' dp '), 2with at most quadratic growth, so that a conservative flow(q) over dot = del V-p + integral del W-p mu, (p) over dot = -del V-q - integral del W-q(mu) is uniquely defined.The dissipative solution is defined by requiring that the equation of p is replaced byp(t) = P-t (p(0) + (0)integral(t)qW mu ds 0 where Pt is the projection on the space of functions corresponding to the restriction map Tt gamma = gamma 1Is>t.Equivalently the particles merge preserving the average momentum p.We obtain several results on the structure of dissipative solutions; among them, regularity, dissipation of energy, approximations with finite particles solutions, density of conservative solutions. The proofs require additional technical difficulties, not present in the analysis of [5] where H(q, p) = p(2)/2.