We study systems of conservation laws arising in two models of adhesion particle dynamics. The first is the system of free particles which stick under collision. The second is a system of gravitationally interacting particles which also stick under collision. In both cases, mass and momentum are conserved at the collisions, so the dynamics is described by 2 x 2 systems of conservations laws. We show that for these systems, global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the Lax-Oleinik variational principle for scalar conservation laws. However, this weak solution is not unique among weak solutions satisfying the standard entropy condition. 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 solutions. We then study the qualitative behavior of the solutions with random initial data. We show that for continuous but nowhere differentiable random initial velocities, all masses immediately concentrate on points even though they were continuously distributed initially, and the set of shock locations is dense.

We first present a new sticky particle method for the system of pressureless gas dynamics. The method is based on the idea of sticky particles, which seems to work perfectly well for the models with point mass concentrations and strong singularity formations. In this method, the solution is sought in the form of a linear combination of δ-functions, whose positions and coefficients represent locations, masses, and momenta of the particles, respectively. The locations of the particles are then evolved in time according to a system of ODEs, obtained from a weak formulation of the system of PDEs. The particle velocities are approximated in a special way using global conservative piecewise polynomial reconstruction technique over an auxiliary Cartesian mesh. This velocities correction procedure leads to a desired interaction between the particles and hence to clustering of particles at the singularities followed by the merger of the clustered particles into a new particle located at their center of mass. The proposed sticky particle method is then analytically studied. We show that our particle approximation satisfies the original system of pressureless gas dynamics in a weak sense, but only within a certain residual, which is rigorously estimated. We also explain why the relevant errors should diminish as the total number of particles increases. Finally, we numerically test our new sticky particle method on a variety of one-and two-dimensional problems as well as compare the obtained results with those computed by a high-resolution finite-volume scheme. Our simulations demonstrate the superiority of the results obtained by the sticky particle method that accurately tracks the evolution of developing discontinuities and does not smear the developing δ-shocks.

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