Analysis of the adhesion model and the reconstruction problem in cosmology
In cosmology, a basic explanation of the observed concentration of mass in singular structures is provided by the Zeldovich approximation, which takes the form of free-streaming flow for perturbations of a uniform Einstein-de Sitter universe in co-moving coordinates. The adhesion model suppresses multi-streaming by introducing viscosity. We study mass flow in this model by analysis of Lagrangian advection in the zero-viscosity limit. Under mild conditions, we show that a unique limiting Lagrangian semi-flow exists. Limiting particle paths stick together after collision and are characterized uniquely by a differential inclusion. The absolutely continuous part of the mass measure agrees with that of a Monge-Ampère measure arising by convexification of the free-streaming velocity potential. But the singular parts of these measures can differ when flows along singular structures merge, as shown by analysis of a 2D Riemann problem. The use of Monge-Ampère measures and optimal transport theory for the reconstruction of inverse Lagrangian maps in cosmology was introduced in work of Brenier & Frisch et al. (Month. Not. Roy. Ast. Soc. 346, 2003). In a neighborhood of merging singular structures in our examples, however, we show that reconstruction yielding a monotone Lagrangian map cannot be exact a.e., even off of the singularities themselves.