Blowups for a class of second order nonlinear hyperbolic equations: A reduced model of nonlinear Jeans instability

TL;DR

This paper studies a simplified model of nonlinear Jeans instability in an expanding universe, establishing blowup solutions and growth estimates that could explain rapid structure formation observed in astrophysics.

Contribution

It introduces a reduced hyperbolic PDE model for nonlinear Jeans instability and constructs stable blowup solutions with growth estimates, advancing understanding of cosmic structure formation.

Findings

Established a family of nonlinear blowup solutions.

Provided growth rate estimates for the solutions.

Suggested explanations for rapid structure formation in the universe.

Abstract

Understanding the formation of nonlinear structures in the universe and stellar systems is crucial. The nonlinear Jeans instability plays a key role in these formation processes. It has been a long-standing open problem in astrophysics for more than a century. In this article, we focus on a reduced model of the nonlinear Jeans instability in an expanding Newtonian universe, which is described by a class of second-order nonlinear hyperbolic equations. \begin{equation*} \Box \varrho(x^\mu) +\frac{\mathcal{a} }{t} \partial_{t}\varrho(x^\mu) - \frac{\mathcal{b}}{t^2} \varrho(x^\mu) (1+ \varrho(x^\mu) ) -\frac{\mathcal{c}-\mathcal{k} }{1+\varrho(x^\mu)} (\partial_{t}\varrho(x^\mu))^2= \mathcal{k} F(t). \end{equation*} We establish a family of nonlinear self-increasing blowup solutions (where the solution itself becomes infinite in a stable ODE-type blowup) for this equation. Furthermore, we provide estimates on the growth rate of $\varrho$, which may help explain why the nonlinear structures in the universe grow much faster in astrophysical observations than predicted by the classical Jeans instability.