Long time dynamics close to large amplitude quasi-periodic traveling waves in two dimensional forced rotating fluids

Abstract: In this paper we consider the $β$-plane equation with a smooth external force which is a quasi-periodic traveling wave of large amplitude $O(λ^{α- 1})$, $1 < α< 2$, and with large speed of propagation of size $O(λ)$. In a previous paper, the second and the third author proved the existence of quasi-periodic traveling wave solutions of large amplitude of order $O(λ<sup>θ)$, for some $θ&gt; 0$. The purpose of this paper is to analyze the long time dynamics for smooth initial data close to these traveling wave solutions. In particular, we shall prove that, for initial data sufficiently close to a fixed traveling wave solution (in the $H<sup>s$ topology), the corresponding solution remains close to the traveling wave solution for arbitrary long time (independent of the size of the traveling wave solution). As a consequence, we prove that there are open sets of large initial for which one has almost global existence, namely such that the corresponding solution remains of the same size of the initial datum for arbitrary long time (independent of the size of the initial data). The proof combines several ingredients: an analysis of the linearized PDE at any traveling wave solution via normal form methods, a sharp analysis of the transformed nonlinear problem under the change of coordinates that diagonalizes the linearized equation and energy estimates.

• The paper proves almost-global nonlinear stability for large amplitude quasi-periodic traveling waves in 2D forced rotating fluids using a multi-step normal form and reducibility analysis.
• It overcomes small divisor and resonance challenges by implementing tailored transformations that yield sharp energy estimates and uniform-in-size bounds.
• The approach extends traditional perturbative methods to control genuinely large nonlinear structures, offering new insights into invariant-like organizing dynamics.

Long-Time Dynamics Near Large Amplitude Quasi-Periodic Traveling Waves in 2D Forced Rotating Fluids

Introduction and Problem Formulation

The paper addresses the long-time nonlinear dynamics of the two-dimensional $\beta$-plane equation

$$\partial_t v + u \cdot \nabla v - \beta\, \mathcal{L} v = {\bf F}(t, x),$$

where $v$ is the scalar vorticity on $\mathbb{T}^2$, $u = \nabla^\perp(-\Delta)^{-1}v$ is the velocity (via Biot–Savart), $\mathcal{L}$ is the dispersive operator $\partial_{x_1}(-\Delta)^{-1}$, and $\beta\neq 0$ encodes the Coriolis effect. The external force ${\bf F}(t,x)$ is assumed to be a quasi-periodic traveling wave of the form $\lambda^\alpha f(\lambda \omega t, x)$ with $$\partial_t v + u \cdot \nabla v - \beta\, \mathcal{L} v = {\bf F}(t, x),$$0, $$\partial_t v + u \cdot \nabla v - \beta\, \mathcal{L} v = {\bf F}(t, x),$$1, and propagation speed $$\partial_t v + u \cdot \nabla v - \beta\, \mathcal{L} v = {\bf F}(t, x),$$2 as $$\partial_t v + u \cdot \nabla v - \beta\, \mathcal{L} v = {\bf F}(t, x),$$3.

Previously, the existence of large amplitude quasi-periodic traveling wave solutions with amplitude $$\partial_t v + u \cdot \nabla v - \beta\, \mathcal{L} v = {\bf F}(t, x),$$4 had been established. This work advances by proving long-time (almost global) nonlinear stability in Sobolev spaces for smooth data near such large amplitude waves, for timescales independent of their size. The regime is genuinely nonperturbative, as the solutions and considered data are not small.

Main Results and Claims

Let $$\partial_t v + u \cdot \nabla v - \beta\, \mathcal{L} v = {\bf F}(t, x),$$5 be a quasi-periodic traveling wave solution with large amplitude of order $$\partial_t v + u \cdot \nabla v - \beta\, \mathcal{L} v = {\bf F}(t, x),$$6. The central results are:

• For any $$\partial_t v + u \cdot \nabla v - \beta\, \mathcal{L} v = {\bf F}(t, x),$$7, and initial data $$\partial_t v + u \cdot \nabla v - \beta\, \mathcal{L} v = {\bf F}(t, x),$$8 with $$\partial_t v + u \cdot \nabla v - \beta\, \mathcal{L} v = {\bf F}(t, x),$$9 (with $v$0 arbitrary), the solution $v$1 to the forced $v$2-plane equation remains within $v$3 in $v$4 of the wave profile for a time interval $v$5, where $v$6, and crucially, $v$7 does not depend on $v$8 or the size of the solution. Furthermore, one can construct open sets of large data in $v$9 (with norm $\mathbb{T}^2$0) for which the solution remains uniformly bounded (in terms of the initial data size) for arbitrarily long times.
• The method yields almost global existence and uniform-in-size bounds for large solutions emanating from data in a ball centered at a traveling wave in $\mathbb{T}^2$1, for arbitrary $\mathbb{T}^2$2.
• The estimates above are sharp: for generic smooth data, the norm can possibly grow double-exponentially in time, as previously shown in the literature.

Technical Architecture

Linearization, Normal Forms, and Reducibility

A pivotal aspect is the detailed analysis of the linearized equation at a large amplitude quasi-periodic wave, for which classical energy-based local theory is inapplicable due to the large size and quasi-linear character. The main difficulties are:

• Small Divisors and Resonances: The linearization leads to time-dependent pseudo-differential operators with bad unbounded perturbations and small divisor phenomena associated to the non-resonant matching of time and space frequencies.
• Large Nonlinear Background: All relevant quantities scale with large powers of $\mathbb{T}^2$3, so standard perturbation theory does not apply.

To overcome this, the authors implement a multi-step normal form and reducibility scheme adapted to the high-degeneracy of the (anisotropic) dispersive operator, leveraging both the traveling wave (momentum-conserving) structure and Diophantine non-resonance for the quasi-periodic frequency vector. The steps are:

1. Straightening the Transport: Remove the quasi-periodic dependence from the linear transport by an $\mathbb{T}^2$4-diffeomorphism, using Diophantine conditions to solve appropriate cohomological equations for the change of variables.
2. Normal Form/Eliasson-Kuksin-type Iteration: Applying a finite sequence of time-dependent invertible transformations (exponentials of smoothing operators), the linear operator is diagonalized up to a rapidly decaying remainder. The homological equations in each step are solvable on large measure sets of parameter values due to conservation laws and non-resonance structure.
3. Quantitative Resolvent and Spectral Analysis: The final normal form is an almost diagonal operator, and the full reducibility/KAM step (for small, rapidly decaying perturbations) can be completed via a standard Nash–Moser/KAM-like iteration to obtain a conjugacy with the diagonal flow.

Crucially, the small divisor structure is tightly controlled by the momentum-preserving property of all operators arising from the traveling wave geometry—this is essential to avoid catastrophic resonances in high dimension.

Reduction of the Nonlinear Problem

The solution is recast as $\mathbb{T}^2$5, where $\mathbb{T}^2$6 solves a perturbed nonlinear hyperbolic system with all "dangerous" time dependence shifted to the linear and quadratic terms. The previous diagonalization yields:

• After all changes of variables, the equation for the renormalized perturbation $\mathbb{T}^2$7 takes the form

$\mathbb{T}^2$8

with $\mathbb{T}^2$9 diagonal with pure imaginary spectrum (hence unitary), and $u = \nabla^\perp(-\Delta)^{-1}v$0 is a nonlinear transport operator admitting sharp energy estimates due to the structure of the change of variables.

• The dominant nonlinear term is of transport type, with commutator structure carefully tracked; all higher-order remainders are smoothing and can thus be handled in $u = \nabla^\perp(-\Delta)^{-1}v$1 by energy methods.
• The transformed nonlinearity, despite the underlying quasi-periodic time dependence, has tame estimates essential for robust control over long timescales (of order $u = \nabla^\perp(-\Delta)^{-1}v$2, for initial perturbation of size $u = \nabla^\perp(-\Delta)^{-1}v$3).

Key Analytical Bounds and Their Role

• Time Scale Independence: All constants in the stability bounds—particularly, the time for which the solution remains controlled—are uniform in $u = \nabla^\perp(-\Delta)^{-1}v$4, i.e., do not deteriorate as the size of the solution increases.
• Nonlinear Stability in Large Regime: The method provides nonlinear stability for large amplitude quasi-periodic waves, showing a lack of energy cascade on arbitrarily long (but not infinite) timescales, for data taken in a sufficiently small neighborhood of the wave profile.
• The size of the neighborhood may need higher regularity compatibility, due to losses from the quasi-linear structure and the small divisor analysis, but is arbitrary (can be made as small as desired).

Implications and Relation to Previous Work

This work stands in contrast to the established works where long-time stability could only be managed for solutions small in Sobolev norm, or only at the linear or weakly nonlinear level. Here, the key advance is the ability to handle quasi-linear, non-small, high-dimensional quasi-periodic structures in a two-dimensional fluid problem, using the fine structure of the PDE and the momentum conservation laws.

The analysis uses and generalizes reducibility and normal form techniques common in the KAM/Nash–Moser theory for NLW/NLS/KdV (see e.g., [Berti-Kappeler-Montalto], [Maspero-Procesi] for NLS/KdV, and [Haus et al.] for 2D NLS), but addresses new challenges inherent to the quasi-periodic, large amplitude, and quasi-linear context in fluid mechanics. The small divisor issues here are more severe due to the low regularity of the dispersion relation and the strong coupling of time and space oscillations.

Among conceptual implications:

• Stability of nontrivial, high-energy coherent structures (e.g., these forced large amplitude traveling waves) for long—though not infinite—time, providing partial positive evidence for the "simple states" conjecture for 2D incompressible fluids.
• The diagonalization and control of the linearized operator modulo large remainders may open the door to further results on even longer time dynamics, or, possibly, on global regularity under more restrictive structural assumptions.
• The technical approach here could inform future work on large amplitude periodic or quasi-periodic solutions in higher dimensions, in other fluid or geostrophic PDEs.
• From a more dynamical systems perspective, the results suggest the existence of large open sets in function space for which invariant-like structures provide organizing centers for long-term dynamics, even outside the perturbative regime.

Conclusion

The paper establishes almost-global nonlinear stability for large amplitude quasi-periodic traveling waves for the forced 2D $u = \nabla^\perp(-\Delta)^{-1}v$5-plane equation, on arbitrarily long timescales independent of amplitude. By combining a fine reducibility analysis (normal forms up to infinite order), resolution of small divisor resonances, and sharp control of the transport nonlinearity, the authors obtain stability results that go substantially beyond conventional perturbative or small-data regimes. These results significantly extend the theoretical understanding of high-dimensional quasi-periodic structures and their robustness in strongly nonlinear fluid evolutions (2604.09302).