Applied Mathematics Colloquium by Jian-Guo Liu: Analysis of the Adhesion Model and Reconstruction in Cosmology
Speaker: , professor of mathematics and physics, Duke University
Title: Analysis of the Adhesion Model and Reconstruction in Cosmology
Abstract: 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 satisfies a Monge-Ampère equation related to convexification of the free-streaming velocity potential.
The use of Monge-Ampère equations and optimal transport theory for the reconstruction of inverse Lagrangian maps in cosmology was introduced in work of Brenier and Frisch et al (2003). We show that the singular part of the mass measure can differ from the Alexandrov solution to the Monge-Ampère equation, however, when flows along singular structures merge, as shown by analysis of a 2D Riemann problem. In a neighborhood of merging singular structures in our examples, we show that reconstruction yielding a monotone Lagrangian map cannot be exact a.e., even off of the singularities themselves.
uniquely by a differential inclusion. The absolutely continuous part of the mass measure satisfies a Monge-Ampère equation related to convexification of the free-streaming velocity potential.
The use of Monge-Ampère equations and optimal transport theory for the reconstruction of inverse Lagrangian maps in cosmology was introduced in work of Brenier and Frisch et al (2003). We show that the singular part of the mass measure can differ from the Alexandrov solution to the Monge-Ampère equation, however, when flows along singular structures merge, as shown by analysis of a 2D Riemann problem. In a neighborhood of merging singular structures in our examples, we show that reconstruction yielding a monotone Lagrangian map cannot be exact a.e., even off of the singularities themselves.
Applied Mathematics Colloquium