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On the chaotic behavior of the Lagrangian flow of the 2D Navier-Stokes system with bounded degenerate noise

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arxiv 2406.17612 v2 pith:QRA7T3QN submitted 2024-06-25 math.AP math.DSmath.PR

On the chaotic behavior of the Lagrangian flow of the 2D Navier-Stokes system with bounded degenerate noise

Vahagn Nersesyan , Deng Zhang , Chenwan Zhou This is my paper
classification math.AP math.DSmath.PR
keywords systembehaviorboundedchaoticexponentflowfluidlagrangian
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We consider a fluid governed by the randomly forced 2D Navier-Stokes system. It is assumed that the force is bounded, acts directly only on a small number of Fourier modes, and satisfies some natural decomposability and observability properties. Under these assumptions, we show that the Lagrangian flow associated with the random fluid exhibits chaotic behavior characterized by the strict positivity of the top Lyapunov exponent. To achieve this, we introduce a new abstract result that allows to derive positivity of the top Lyapunov exponent from controllability properties of the underlying deterministic system.

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Cited by 4 Pith papers

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    Weakly damped KdV on a circle with bounded localized random forcing is exponentially mixing: a unique invariant measure exists and attracts all initial data in L2.

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    The 1D stochastic Allen-Cahn equation with localized white noise admits a unique invariant measure and its Markov process is exponentially mixing.

  3. Exponential mixing for nonlinear Schr\"odinger equations perturbed by bounded degenerate noise

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    Exponential mixing to a unique invariant measure is established for locally damped NLS with bounded degenerate noise on two modes using a new criterion based on asymptotic compactness of the linearized system.

  4. Lagrangian chaos for the 2D Navier-Stokes equations driven by mildly degenerate noise

    math.DS 2026-03 unverdicted novelty 7.0

    The top Lyapunov exponent of the Lagrangian flow for 2D incompressible Navier-Stokes equations driven by mildly degenerate noise on low modes is strictly positive.