2024/09/04 by Liu, Chao · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2409.02516
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: ∂2t \varrho- \biggl(\frac m2 (∂t\varrho )2(1+\varrho )2 + 4(k-m2)(1+\varrho )\biggr) Δ\varrho = F(t,\varrho,∂μ \varrho) where F is given by F(t,\varrho,∂μ \varrho):= \underbrace(2)/(3 ) \varrho (1+\varrho) (i) self-increasing \underbrace-(1)/(3) ∂t\varrho (ii) damping + \underbrace(4)/(3) \frac(∂t\varrho )21+\varrho (iii) Riccati + \underbrace \biggl(m2 \frac (∂t\varrho )2(1+\varrho )2 + 4(k-m2) (1+\varrho ) \biggr) qi ∂i\varrho (iv) convection - \mathttKij ∂i\varrho∂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.