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 \in \mathscr{P}( \mathbb {R}^d \times \mathbb {R}^d) $ in the sense of [3]. The Hamiltonian is of the form

$ \begin{equation*} H(\mu) = \int V(q,p) \mu(dqdp) + \frac{1}{2} \int \int W(q,p,q',p') \mu(dqdp)\mu(dq'dp'), \end{equation*} $

with at most quadratic growth, so that a conservative flow

$ \begin{equation*} \dot q = \nabla_p V + \int \nabla_p W \mu, \quad \dot p = - \nabla_q V - \int \nabla_q W \mu \end{equation*} $

is uniquely defined.

The dissipative solution is defined by requiring that the equation of $ p $ is replaced by

$ \begin{equation*} p(t) = \mathbb{P}_t \bigg( p(0) + \int_0^t \bigg[ - \nabla_q V - \int \nabla_q W \mu \bigg] ds \bigg). \end{equation*} $

where $ \mathbb P_t $ is the projection on the space of functions corresponding to the restriction map

$ \mathbb T_t \gamma = \gamma 1\!\!\mathrm{I}_{s > 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 $.