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The emergence of nonlinear Jeans-type instabilities for quasilinear wave equations. II: Generalizations

Chao Liu, Yiqing Shi

TL;DR

This work extends nonlinear Jeans-type instability analysis to a broad family of quasilinear wave equations by coupling a reference ODE profile with a PDE through a two-stage time-compactification and a Fuchsian reformulation. The authors establish that small long-wavelength perturbations lead to self-growth blow-up along null geodesics, with a well-characterized blow-up endpoint $p_m$ and precise bounds in both inhomogeneous and homogeneous regions. Central to the approach are the time compactifications synchronizing the reference and perturbed solutions, a zoom-in coordinate framework, and the lens-shaped region analysis which enables a Fuchsian global initial-value problem on a compact manifold $\mathbb{T}^n$. The results generalize prior work to parameter ranges $1<\mathsf{a}\leq 30$, $\tfrac{1}{3}\leq\mathsf{b}\leq\tfrac{2}{3}$ (with $\mathsf{c}=\tfrac{4}{3}$), providing sharp growth estimates and region-by-region dynamics relevant to nonlinear Jeans-type instabilities and their potential implications for structure formation models.

Abstract

This work extends the previous work by the first author [arXiv:2409.02516] and [Math. Ann. 393 (2025), 317-363], analyzing the long-term behavior of solutions to a broader class of quasilinear wave equations with parameter $1<\mathsf{a}\leq30$ and $\frac{1}{3}\leq\mathsf{b}\leq\frac{2}{3}$: \begin{equation*} \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) Δ\varrho = F(t,\varrho,\partial_μ \varrho) \end{equation*} where $F$ is given by \begin{equation*} F(t,\varrho,\partial_μ \varrho):= \mathsf{b} \varrho (1+ \varrho ) -(\mathsf{a}-1) \partial_{t}\varrho + \frac{4}{3} \frac{(\partial_{t}\varrho )^2}{1+\varrho } + \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 - \mathtt{K}^{ij} \partial_{i}\varrho\partial_{j}\varrho . \end{equation*} The results demonstrate that for this extensive family of quasilinear wave equations satisfying $1<\mathsf{a}\leq30$ and $\frac{1}{3}\leq\mathsf{b}\leq\frac{2}{3}$, self-increasing blowup solutions also exist, and self-increasing singularities emerge at certain future endpoints of null geodesics provided the inhomogeneous perturbations of data are sufficiently small.

The emergence of nonlinear Jeans-type instabilities for quasilinear wave equations. II: Generalizations

TL;DR

This work extends nonlinear Jeans-type instability analysis to a broad family of quasilinear wave equations by coupling a reference ODE profile with a PDE through a two-stage time-compactification and a Fuchsian reformulation. The authors establish that small long-wavelength perturbations lead to self-growth blow-up along null geodesics, with a well-characterized blow-up endpoint and precise bounds in both inhomogeneous and homogeneous regions. Central to the approach are the time compactifications synchronizing the reference and perturbed solutions, a zoom-in coordinate framework, and the lens-shaped region analysis which enables a Fuchsian global initial-value problem on a compact manifold . The results generalize prior work to parameter ranges , (with ), providing sharp growth estimates and region-by-region dynamics relevant to nonlinear Jeans-type instabilities and their potential implications for structure formation models.

Abstract

This work extends the previous work by the first author [arXiv:2409.02516] and [Math. Ann. 393 (2025), 317-363], analyzing the long-term behavior of solutions to a broader class of quasilinear wave equations with parameter and : \begin{equation*} \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) Δ\varrho = F(t,\varrho,\partial_μ \varrho) \end{equation*} where is given by \begin{equation*} F(t,\varrho,\partial_μ \varrho):= \mathsf{b} \varrho (1+ \varrho ) -(\mathsf{a}-1) \partial_{t}\varrho + \frac{4}{3} \frac{(\partial_{t}\varrho )^2}{1+\varrho } + \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 - \mathtt{K}^{ij} \partial_{i}\varrho\partial_{j}\varrho . \end{equation*} The results demonstrate that for this extensive family of quasilinear wave equations satisfying and , self-increasing blowup solutions also exist, and self-increasing singularities emerge at certain future endpoints of null geodesics provided the inhomogeneous perturbations of data are sufficiently small.

Paper Structure

This paper contains 34 sections, 31 theorems, 178 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Methodology, Assumptions and Main Theorems
  3. Basic Assumptions and Main Theorems
  4. Methods and Ideas
  5. Summaries and outlines
  6. Notations
  7. Indices and coordinates
  8. Remainder terms
  9. Derivatives
  10. Function spaces and matrix inequalities
  11. Compactified time transformations and analysis of the reference equation
  12. Time Compactification for the Reference Solution
  13. Time Compactification for the Perturbed Solution
  14. Singular Symmetric Hyperbolic System
  15. Derivation of the Singular Form
  16. ...and 19 more sections

Key Result

Theorem 1.1

For Eq1, assume $k \in \mathbb{Z}_{\geq \frac{n}{2} + 3}$, the parameters $\mathsf{a}, \mathsf{b}, \mathsf{c}$ satisfy condition A:2, the constants $\mathtt{A}, \mathtt{B}, \mathtt{C}, \mathtt{D}$ defined in e:ttA--e:ttD depend on the initial values $\beta$ and $\beta_0$, and Assumptions A:1--A:4 ho then there exists a hypersurface $t = \mathcal{T}(x, \delta_0)$ satisfying so that Eq1 admits a so

Figures (4)

  • Figure 1: Regions for the solution in the main theorems.
  • Figure 2: The compactified coordinate system $(\tau,\zeta)$ after time compactification.
  • Figure 3: The zoom-in coordinate system $(\tilde{\tau},\tilde{\zeta})$.
  • Figure 4: The region $\widehat{\mathit{\Lambda}_{\delta_0}}$ and the characteristic surface $\widehat{\mathcal{C}}$

Theorems & Definitions (46)

  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • ...and 36 more