[Paper Review] The Inverse Problem for the Euler-Poisson system in Cosmology

Grégoire Loeper|ArXiv.org|Jun 30, 2003

Geometric Analysis and Curvature Flows15 references3 citations

TL;DR

This paper formulates the cosmological reconstruction problem for the Euler-Poisson system as a convex minimization of the Lagrangian action, establishing a variational framework analogous to optimal transport. It proves the existence of weak solutions that are consistent with smooth solutions and exhibit log-Lipschitz regularity of the velocity potential on the support of the density, ensuring structural stability and reversibility of the dynamics on the support of the measure.

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.

Motivation & Objective

• To address the ill-posed inverse problem of reconstructing intermediate cosmological states from initial and final density fields.
• To reformulate the reconstruction problem as a convex minimization of the action functional, leveraging variational principles.
• To establish existence and regularity of weak solutions under the slaving condition (velocity as gradient of a potential).
• To demonstrate that minimizing solutions are consistent with smooth solutions of the Euler-Poisson system.
• To derive log-Lipschitz regularity of the velocity potential on the support of the density, ensuring structural stability.

Proposed method

• Formulates the Euler-Poisson system on the flat torus with a neutralizing background, reducing Poisson's equation to Δp = ρ − 1.
• Replaces the Cauchy problem with a two-point boundary value problem, prescribing initial and final densities ρ₀ and ρₜ.
• Replaces the dynamics with a variational principle minimizing the action functional I(ρ, v, p) = ∫₀ᵀ ∫|v|² dρ + |∇p|² dx dt.
• Applies Monge-Kantorovitch duality from optimal transport theory to analyze the minimization problem.
• Uses viscosity solutions of the Hamilton-Jacobi equation ∂ₜφ + ½|∇φ|² + p = 0 to characterize the velocity potential φ.
• Establishes that the solution φ is differentiable and log-Lipschitz continuous on the support of ρ, with uniform bounds in time.

Experimental results

Research questions

• RQ1 Can the intermediate evolution of a self-gravitating fluid be reconstructed from only the initial and final density fields?
• RQ2 Does the variational formulation of the Euler-Poisson system via action minimization yield solutions consistent with classical smooth solutions?
• RQ3 What regularity properties does the velocity potential φ inherit on the support of the density ρ in the minimizing solution?
• RQ4 How does the structure of the optimal transport framework apply to the cosmological reconstruction problem?
• RQ5 Is the dynamics reversible on the support of the density in the minimizing solution?

Key findings

• Weak solutions to the Euler-Poisson system exist as minimizers of the action functional under the slaving condition, ensuring well-posedness of the reconstruction problem.
• The minimizing solution satisfies the Hamilton-Jacobi equation ∂ₜφ + ½|∇φ|² + p = 0, with φ being the velocity potential.
• On the support of ρ(t), the velocity potential φ is differentiable and satisfies a log-Lipschitz continuity condition: |∇φ(t,x) − ∇φ(t,y)| ≤ C(τ)|x−y| log(1/|x−y|) for t ∈ [τ, T−τ].
• The solution is reversible on the support of ρ, meaning the forward and backward dynamics are consistent, a property not generally true for viscosity solutions.
• If initial and final densities ρ₀, ρₜ are in Lᵏ(𝕋ᵈ) with k > d, then the velocity potential φ belongs to W¹,∞([0,T]×𝕋ᵈ), ensuring bounded gradients.
• The regularity of p and φ is preserved uniformly in time away from the endpoints, with p ∈ C¹,LlogL and φ ∈ C¹,LlogL on compact subintervals of (0,T).