The Inverse Problem for the Euler-Poisson system in Cosmology

TL;DR

This paper investigates the inverse problem for the Euler-Poisson system in cosmology, establishing existence of weak solutions via variational methods, exploring their regularity, and connecting to Hamilton-Jacobi equations.

Contribution

It introduces a novel variational approach to solve the inverse Euler-Poisson problem and analyzes the regularity and connections to Hamilton-Jacobi equations.

Findings

Existence of weak solutions via action minimization.

Solutions are consistent with smooth solutions and have special regularity.

Links established between Euler-Poisson solutions and Hamilton-Jacobi equations.

Abstract

The motion of a continuum of matter subject to gravitational interaction is classically described by the Euler-Poisson system. Prescribing the density of matter at initial and final times, we are able to obtain weak solutions for this equation by minimizing the action of the Lagrangian which is a convex functional. Then we see that such minimizing solutions are consistent with smooth solutions of the Euler-Poisson system and enjoy some special regularity properties. Meanwhile some intersting links with with Hamilton-Jacobi equations are found.