Rigorous proof of slightly nonlinear Jeans instability in the expanding Newtonian universe

TL;DR

This paper develops a novel non-Fourier analytical method to rigorously prove the slightly nonlinear Jeans instability in an expanding Newtonian universe, extending classical linear results to nonlinear regimes.

Contribution

It introduces a new approach based on Fuchsian systems to analyze nonlinear Jeans instability, overcoming limitations of Fourier analysis in nonlinear settings.

Findings

Established a rigorous proof for linearized Jeans instability in nonlinear context.

Extended the proof to slightly nonlinear Euler-Poisson equations.

Provided a new mathematical framework for nonlinear gravitational instability analysis.

Abstract

Due to the nonlinearity of the Euler{Poisson equations, it is possible that the nonlinear Jeans instability may lead to a faster density growing rate than the rate in the standard theory of linearized Jeans instability, which motivates us to study the nonlinear Jeans instability. The aim of this article is to develop a method proving the Jeans instability for slightly nonlinear Euler-Poisson equations in the expanding Newtonian universe. The standard proofs of the Jeans instability rely on the Fourier analysis. However, it is difficult to generalize Fourier method to a nonlinear setting, and thus there is no result in the nonlinear analysis of Jeans instability. We firstly develop a non-Fourier-based method to reprove the linearized Jeans instability in the expanding Newtonian universe. Secondly, we generalize this idea to a slightly nonlinear case. This method relies on the Cauchy problem of the Fuchsian system due to the recent developments of this system in mathematics. The fully nonlinear Jeans instability for the Euler-Poisson and Einstein-Euler equations are in progress.