Lower bounds on the top Lyapunov exponent for linear PDEs driven by the 2D stochastic Navier-Stokes equations
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909391797616640 |
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| author | Hairer, Martin Punshon-Smith, Sam Rosati, Tommaso Yi, Jaeyun |
| author_facet | Hairer, Martin Punshon-Smith, Sam Rosati, Tommaso Yi, Jaeyun |
| contents | We consider the top Lyapunov exponent associated to the advection-diffusion and linearised Navier-Stokes equations on the two-dimensional torus. The velocity field is given by the stochastic Navier-Stokes equations driven by a non-degenerate white-in-time noise with a power-law correlation structure. We show that the top Lyapunov exponent is bounded from below by a negative power of the diffusion parameter. This partially answers a conjecture of Doering and Miles and provides a first lower bound on the Batchelor scale in terms of the diffusivity. The proof relies on a robust analysis of the projective process associated to the linear equation, through its spectral median dynamics. We introduce a probabilistic argument to show that high-frequency states for the projective process are unstable under stochastic perturbations, leading to a Lyapunov drift condition and quantitative-in-diffusivity estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_10419 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower bounds on the top Lyapunov exponent for linear PDEs driven by the 2D stochastic Navier-Stokes equations Hairer, Martin Punshon-Smith, Sam Rosati, Tommaso Yi, Jaeyun Probability Analysis of PDEs Dynamical Systems 60H15, 35Q35, 37H15, 37L30 We consider the top Lyapunov exponent associated to the advection-diffusion and linearised Navier-Stokes equations on the two-dimensional torus. The velocity field is given by the stochastic Navier-Stokes equations driven by a non-degenerate white-in-time noise with a power-law correlation structure. We show that the top Lyapunov exponent is bounded from below by a negative power of the diffusion parameter. This partially answers a conjecture of Doering and Miles and provides a first lower bound on the Batchelor scale in terms of the diffusivity. The proof relies on a robust analysis of the projective process associated to the linear equation, through its spectral median dynamics. We introduce a probabilistic argument to show that high-frequency states for the projective process are unstable under stochastic perturbations, leading to a Lyapunov drift condition and quantitative-in-diffusivity estimates. |
| title | Lower bounds on the top Lyapunov exponent for linear PDEs driven by the 2D stochastic Navier-Stokes equations |
| topic | Probability Analysis of PDEs Dynamical Systems 60H15, 35Q35, 37H15, 37L30 |
| url | https://arxiv.org/abs/2411.10419 |