Matter evolution in Burgulence

Abstract

In inviscid solutions of the forced Burgers equation the matter accumulates in the shock discontinuities. We describe the limit motion of particles everywhere including the shocks as the trajectories of a discontinuous velocity field being a generalization of the gradient of the limit potential. The latter is not differentiable but satisfies some convexity properties which guarantee the existence of the gradient. It turns out that for such discontinuous gradient ordinary differential equations there are natural existence, uniqueness, and continuity theorems. The general results are applied for investigation of formation and motion in plane of massive points which are interpreted as various clusters in the adhesion model of the Universe.