Nonlinear Instability of Roll Solutions

• The paper demonstrates that finite-amplitude perturbations can trigger sustained nonlinear growth in roll solutions, confirming that spectral instability leads to significant deviation from equilibrium.
• It uses Bloch-Floquet decomposition and semigroup bounds to quantify critical transition times and amplitude separation, highlighting explicit metrics such as O(|ln δ|) scaling.
• The analysis identifies bifurcation structures, cubic nonlinear saturation, and robust nonlinear damping mechanisms as key insights for understanding roll instability and pattern transitions.

Nonlinear instability of roll solutions refers to the breakdown or destabilization, under finite-amplitude perturbations, of periodic or axisymmetric flow structures characterized by spatially repeating “rolls.” These phenomena are central in a range of pattern-forming systems governed by partial differential equations—including reaction–diffusion models, hydrodynamic equations, and thin film flows—where the transition from linear (spectral) instability to genuinely nonlinear growth underlies complex spatiotemporal behavior such as turbulence, pattern selection, and subcritical chaos.

1. Mathematical Formulation and Representative Roll Solutions

Canonical roll solutions arise as spatially periodic or axisymmetric equilibria of systems such as the generalized Swift–Hohenberg equation (gSHE), Navier–Stokes equations in rotor–stator geometry, and thin-film St. Venant equations. For the gSHE in $\mathbb{R}^2$,

$$\partial_t u = -\left(1 + \partial_x^2 + \partial_y^2\right)^2 u + \varepsilon^2 u + b u^2 - s u^3,$$

small-amplitude rolls are spatially periodic in $x$ and constant in $y$, constructed via Lyapunov–Schmidt reduction for small $\varepsilon$ and detuning parameter $\omega$ (Chae et al., 30 Nov 2025).

In axisymmetric rotor–stator flow,

$$\frac{\partial \vec{u}}{\partial t} + (\vec{u}\cdot\nabla)\vec{u} = -\nabla p + \frac{1}{Re} \nabla^2 \vec{u},\qquad \nabla\cdot\vec{u} = 0,$$

boundary conditions enforce rigid rotation and stator constraints. Rolls manifest as concentric vortices about the axis, localized in Bödewadt boundary layers (Gesla et al., 2023, Gesla et al., 20 Dec 2024).

Roll waves in thin film flow are periodic profiles that balance advection, gravity, and viscosity in the St. Venant system; their detailed structure depends on the Froude number $F$ (Rodrigues et al., 2015).

2. Spectral Instability and Bloch-Floquet Linearization

Linear stability analysis proceeds via Bloch-Floquet decomposition exploiting underlying spatial periodicity or homogeneity:

• For gSHE rolls, the linearized operator $L$ admits a decomposition into Bloch modes $B(\sigma)$ parametrized by wavevector $\sigma\in\mathbb{R}^2$; spectral instability is established via existence of $\sigma^*$ for which $\operatorname{Re}\lambda(\sigma^*)>0$ (Chae et al., 30 Nov 2025).
• In rotor–stator flows, axisymmetric perturbations are expanded in normal modes; the critical Reynolds number $Re_c$ is identified where the leading eigenvalue crosses zero, signaling the onset of roll instability (Gesla et al., 2023, Gesla et al., 20 Dec 2024).
• Thin film roll waves require Floquet analysis of the linearized St. Venant operator; spectral stability conditions (D1–D3) are necessary for nonlinear stability (Rodrigues et al., 2015).

Spectral instability is a prerequisite for transitional growth but does not, by itself, guarantee that finite-amplitude perturbations induce large-scale deviation from the equilibrium.

3. Nonlinear Instability: Rigorous Transition and Growth Mechanism

Nonlinear instability is established by constructing small initial perturbations that, under the full nonlinear dynamics, undergo sustained growth leading to finite departure from the roll solution within logarithmic time. The methodology employed in (Chae et al., 30 Nov 2025) for the 2D gSHE involves:

• Building a perturbation concentrated near the maximally unstable Bloch mode:

$$U(x, y) = \int_{I \times J} e^{i(\sigma_1 x + \sigma_2 y)} W(x, \sigma) d\sigma_1 d\sigma_2$$

• Controlling nonlinear corrections via an iterated approximate solution $V^{\text{app}}(t)$ up to order $N$, then estimating the remainder in $H^2$ via semigroup bounds and energy estimates.
• Proving that for $t = T^\delta = O(|\ln\delta|)$, the solution exits an $O(1)$ neighborhood:

$$\|V_\delta(T^\delta) - \tilde u\|_{L^2(\mathbb{R}^2)} \geq \theta > 0$$

• The precise theorem demonstrates that there are always perturbations (arbitrarily small in $H^4$) that produce large deviations in finite time when spectral instability is present.

A plausible implication is that the nonlinear instability of rolls is not only sufficient to generate pattern transition but is generically realized in multidimensional systems whenever spectral instability is identified.

4. Subcritical Instabilities, Edge States, and Bifurcation Structure

In rotor–stator flows, subcritical branches and chaotic dynamics are traced via advanced bifurcation methods:

• Supercritical Hopf branches observed near $Re_c$ are limited to small amplitude and narrow $Re$ intervals.
• A periodic-subcritical branch, found via harmonic balance methods (HBM), emerges at $Re_{SN} < Re_c$ and supports finite-amplitude periodic solutions disconnected from the base flow. This branch is responsible for experimentally observed rolls at low $Re$ and merges with the edge/chaotic manifold at $Re\approx1800$ (Gesla et al., 2023).
• Edge states, determined by bisection algorithms in phase space, act as separatrices between relaminarization and chaotic roll formation.
• Subcritical chaotic branches and associated hysteresis phenomena are shown to arise in the homotopy deformation from annular cavity to rotor–stator geometry, with modal competition and roll pairing governed by nonlinear resonance (Gesla et al., 20 Dec 2024).

This suggests that the existence of subcritical branches and edge states provides a finite-amplitude route to turbulence and complex roll dynamics even when linear stability persists.

5. Saturation of Instability and Nonlinear Amplitude Equations

Weakly nonlinear expansions yield amplitude equations governing saturation:

• In viscous boundary layers, cubic nonlinear interactions restrict the growth of long-wave instabilities to saturated amplitude $O(\nu^{1/4})$; cubic coefficients are computed by matched asymptotic expansions in the critical layer (Bian et al., 2022).
• For rotor–stator and Couette roll/streak instabilities, Stuart–Landau or Landau equations,

$$\frac{dA}{dt} = \sigma(Re - Re_c)A + \beta|A|^2A + \cdots$$

describe the bifurcation; supercritical branches are confirmed when $\beta<0$ with amplitude scaling $A\sim (Re - Re_c)^{1/2}$ (Gesla et al., 2023).

• In SSST/S3T frameworks, the amplitude equation includes cubic saturation and yields equilibria representing steady roll/streak complexes (Farrell et al., 2010, Farrell et al., 2016).

A plausible implication is that nonlinear saturation mechanisms, determined by cubic and higher-order terms, set quantitative bounds for the amplitude of roll solutions post-instability.

6. Nonlinear Damping, Modulated Energy Estimates, and Stability Theory

Periodic-coefficient damping estimates underpin nonlinear stability analyses in systems such as roll waves in thin films:

• Weighted energies incorporating periodic coefficients yield robust $H^s$ nonlinear damping estimates without requiring pointwise positivity (“slope condition”); instead, an averaged slope condition suffices:

$$\frac{1}{\nu}\langle \alpha \bar\tau^2 \rangle = \langle \bar\tau^{-1} F^{-2} \rangle > 0$$

for all Froude numbers and periodic profiles (Rodrigues et al., 2015).

• High-frequency resolvent bounds and Kawashima-type damping establish energy decay and enable closure of nonlinear stability iterations provided spectral conditions (D1–D3) hold.

This framework confirms that nonlinear instability is only induced if spectral instability is present and rules out supplementary mechanisms associated with algebraic or pointwise coefficient violations.

7. Implications, Limitations, and Open Problems

The rigorous transition from spectral to nonlinear instability across multi-dimensional, periodic, and boundary-driven systems has been established for several canonical PDEs. Notable implications and open directions include:

• The presence of nonlinear instability for arbitrarily small perturbations in multidimensional domains whenever the spectral criterion is satisfied (Chae et al., 30 Nov 2025).
• The saturation of roll amplitude in viscous layers by cubic interactions, raising questions about universality and model-dependent limitations (Bian et al., 2022).
• The bifurcation-theoretic identification of subcritical and supercritical branches, edge states, and chaotic attractors underpins the sensitivity to finite-amplitude disturbances, with applications in turbulence onset and pattern selection (Gesla et al., 2023, Gesla et al., 20 Dec 2024).
• Remaining analytic challenges involve the control of remainder terms in weakly nonlinear expansions, justification of matched asymptotics in complex domains, and a deeper understanding of non-normal mechanisms in turbulence transition.

Future directions will likely focus on refining semigroup techniques for higher-dimensional systems, developing unified bifurcation diagrams across models, and quantifying multiscale interactions in physically relevant boundary conditions.