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On quantitative linear gravitational relaxation

Mahir Hadzic, Matthew Schrecker

TL;DR

This work proves the first quantitative linear decay result for the gravitational Vlasov-Poisson system around nontrivial, compactly supported radially symmetric equilibria with a central point mass. By formulating the problem in action-angle variables and placing a Birkhoff-Poincaré canonical normal form near trapping, the authors overcome stable trapping and derive a limiting absorption principle with uniform resolvent bounds. The decay of the gravitational potential and macroscopic density follows from a delicate combination of pure transport decay, near-resonant/non-resonant frequency splitting, and careful control of Green’s function regularity via a new action-support foliation. The results yield a scattering description toward a nearby linear asymptotic state and provide a robust framework potentially applicable to other inhomogeneous, trapped kinetic systems. The approach highlights how regularity of the steady state and initial data improves decay, aligning with Landau damping heuristics in a gravitational context.

Abstract

We prove quantitative decay rates for the linearised Vlasov-Poisson system around compactly supported equilibria. More precisely, we prove decay of the gravitational potential induced by the radial dynamics of this system in the presence of a point mass source. Our result can be interpreted as the gravitational version of linear Landau damping in the radial setting and hence the first linear asymptotic stability result around such equilibria. We face fundamental obstacles to decay caused by the presence of stable trapping in the problem. To overcome these issues we introduce several new ideas. We use different tools, including the Birkhoff-Poincaré normal form, action-angle type variables, and delicate resolvent bounds to prove a suitable version of the limiting absorption principle and obtain the decay-in-time.

On quantitative linear gravitational relaxation

TL;DR

This work proves the first quantitative linear decay result for the gravitational Vlasov-Poisson system around nontrivial, compactly supported radially symmetric equilibria with a central point mass. By formulating the problem in action-angle variables and placing a Birkhoff-Poincaré canonical normal form near trapping, the authors overcome stable trapping and derive a limiting absorption principle with uniform resolvent bounds. The decay of the gravitational potential and macroscopic density follows from a delicate combination of pure transport decay, near-resonant/non-resonant frequency splitting, and careful control of Green’s function regularity via a new action-support foliation. The results yield a scattering description toward a nearby linear asymptotic state and provide a robust framework potentially applicable to other inhomogeneous, trapped kinetic systems. The approach highlights how regularity of the steady state and initial data improves decay, aligning with Landau damping heuristics in a gravitational context.

Abstract

We prove quantitative decay rates for the linearised Vlasov-Poisson system around compactly supported equilibria. More precisely, we prove decay of the gravitational potential induced by the radial dynamics of this system in the presence of a point mass source. Our result can be interpreted as the gravitational version of linear Landau damping in the radial setting and hence the first linear asymptotic stability result around such equilibria. We face fundamental obstacles to decay caused by the presence of stable trapping in the problem. To overcome these issues we introduce several new ideas. We use different tools, including the Birkhoff-Poincaré normal form, action-angle type variables, and delicate resolvent bounds to prove a suitable version of the limiting absorption principle and obtain the decay-in-time.

Paper Structure

This paper contains 34 sections, 42 theorems, 368 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Steady states
  3. Main result
  4. Strategy of the proof
  5. Action-angle variables and the limiting absorption principle
  6. Action-angle variables and trapping
  7. Energy-momentum support and notation.
  8. The angle coordinate
  9. Period and frequency
  10. Regularity properites
  11. Hamiltonian geometry, initial data, and the spectrum
  12. Frequency formulation and Stone's formula
  13. Function spaces
  14. Canonical Normal Form and a new foliation of the action support
  15. Canonical Normal Form
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $\mu,\nu$ satisfy E:SS2. Then there exists $0<\eta_0\ll1$ such that the following holds. Let $f_\eta^{\mu,\nu}$ be a steady state satisfying E:SS with $\eta\in[0,\eta_0)$. Let $1\leq k\leq N+1$ and $f_0\in C^{k}(\overline{\Omega})$ be an initial datum satisfying the orthogonality condition E:FIN where

Figures (2)

  • Figure 1: Left: the $(E,L)$-support $\mathcal{I}$ of a spherically symmetric steady state. The magenta and green parts of the boundary of $\mathcal{I}$ depict $\partial \mathcal{I}_{0}$ and $\partial \mathcal{I}_{\text{vac}}$, respectively (see \ref{['E:IBDRYDEF']}). Right: schematic depiction of $(y,z)$-level sets for some fixed value of $R\in[R_{\textup{min}},R_{\textup{max}}]$. Blue lines are the $y$-level sets and are almost vertical, the red lines are the $z$-level sets.
  • Figure 2: The $L$-dependent family of effective potentials $\Psi_L$.

Theorems & Definitions (97)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2: Regularity lemma I
  • proof
  • Lemma 2.3: Regularity lemma II
  • proof
  • ...and 87 more