The emergence of nonlinear Jeans-type instabilities for quasilinear wave equations

Abstract

This article contributes a key ingredient to the longstanding open problem of understanding the fully nonlinear version of Jeans instability, as highlighted by A. Rendall [Living Rev. Relativ. 8, 6 (2005)]. We establish a family of self-increasing blowup solutions for the following class of quasilinear wave equations (a model of the Peebles' and Noh-Hwang's equations) that have not previously been studied: \[ \partial^2_t \varrho- \biggl(\frac{ \mathsf{m}^2 (\partial_{t}\varrho )^2}{(1+\varrho )^2} + 4(\mathsf{k}-\mathsf{m}^2)(1+\varrho )\biggr) \Delta \varrho = F(t,\varrho,\partial_{\mu} \varrho) \] where $F$ is given by \[ F(t,\varrho,\partial_{\mu} \varrho):= \underbrace{\frac{2}{3 } \varrho (1+\varrho) }_{ \text{(i) self-increasing}} \underbrace{-\frac{1}{3} \partial_{t}\varrho }_{ \text{(ii) damping}} + \underbrace{\frac{4}{3} \frac{(\partial_{t}\varrho )^2}{1+\varrho } }_{\text{(iii) Riccati}} + \underbrace{ \biggl(\mathsf{m}^2 \frac{ (\partial_{t}\varrho )^2}{(1+\varrho )^2} + 4(\mathsf{k}-\mathsf{m}^2) (1+\varrho ) \biggr) q^i \partial_{i}\varrho }_{\text{(iv) convection}} - \mathtt{K}^{ij} \partial_{i}\varrho\partial_{j}\varrho. \] The result implies the solutions can attain arbitrarily large values over time, leading to self-increasing singularities at some future endpoints of null geodesics provided the inhomogeneous perturbations of data are sufficiently small. Moreover, the solution exhibits almost blowup behavior in the long-wavelength domain. This phenomenon is referred to as the nonlinear Jeans-type instability because this wave equation is closely related to the nonlinear version of the Jeans instability problem in the Euler-Poisson and Einstein-Euler systems, which characterizes the formation of nonlinear structures in the universe. The growth rate of $\varrho$ is significantly faster than that of the solutions to the classical linearized Jeans instability.