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Asymptotic stability of the Kolmogorov flow at high Reynolds numbers

Qi Chen, Hao Jia, Dongyi Wei, Zhifei Zhang

TL;DR

This work establishes the nonlinear asymptotic stability of the Kolmogorov flow for the 2D Navier–Stokes equations on a non-square torus in the high-Reynolds regime. The authors develop a robust quasilinear framework that decomposes perturbations into a carefully constructed linear part $\omega_L$ and a nonlinear remainder $\omega_e$, then control $\omega_e$ via spacetime estimates tied to resolvent bounds for the linearized Euler and NS operators. A key technical breakthrough is the sharp linear-Euler analysis near Kolmogorov flow, including an almost conserved quantity $\omega_{1k}$ and vorticity-depletion effects at critical points, which together enable a nontrivial $\nu^{1/3}$ transition threshold in Sobolev perturbations. The results demonstrate rapid metastable relaxation to a near-Kolmogorov shear flow before slow diffusion-dominated decay to zero, and reveal intricate time-scale structure up to $t\sim 1/\nu$. The methods, combining enhanced dissipation, inviscid damping, and vorticity depletion with resolvent and multiscale analysis, provide a flexible blueprint for nonlinear stability problems in high-Reynolds incompressible flows and may extend to other non-monotone shear profiles.

Abstract

In this paper we prove the asymptotic stability of the Kolmogorov flow on a non-square torus for perturbations $ω_0$ satisfying $\|ω_0\|_{H^3}\llν^{1/3}$, where $0<ν\ll1$ is the viscosity. Kolmogorov flows are important metastable states to the two dimensional incompressible Navier Stokes equations in the high Reynolds number regime. Our result shows that the perturbed solution will rapidly converge to a shear flow close to the Kolmogorov flow, before settling down to the Kolmogorov flow and slowly decaying to $0$ as $t\to\infty$. In fact, our analysis reveals several interesting time scales and rich dynamical behavior of the perturbation in the transition period $0<t\leq 1/ν$. The threshold $ν^{1/3}$, which is the same as that for the Couette flow, is quite surprising since one of the key stability mechanisms, enhanced dissipation, becomes considerably weaker in the case of Kolmogorov flows due to the presence of critical points. To overcome this essential new difficulty, we establish sharp vorticity depletion estimates near the critical points to obtain improved decay rates for the vorticity and velocity fields that are comparable with those for Couette flows, at least for our purposes. We then combine these estimates (enhanced dissipation, inviscid damping and vorticity depletion) with a quasilinear approximation scheme and a multiple-timescale analysis naturally adapted to the dynamics of the perturbation, to obtain the $ν^{1/3}$ threshold for dynamic stability of Kolmogorov flows. The threshold is expected to be sharp when the perturbation is considered in Sobolev spaces. This appears to be the first result that applies vorticity depletion estimates to improve thresholds for nonlinear asymptotic stability in incompressible fluid equations.

Asymptotic stability of the Kolmogorov flow at high Reynolds numbers

TL;DR

This work establishes the nonlinear asymptotic stability of the Kolmogorov flow for the 2D Navier–Stokes equations on a non-square torus in the high-Reynolds regime. The authors develop a robust quasilinear framework that decomposes perturbations into a carefully constructed linear part and a nonlinear remainder , then control via spacetime estimates tied to resolvent bounds for the linearized Euler and NS operators. A key technical breakthrough is the sharp linear-Euler analysis near Kolmogorov flow, including an almost conserved quantity and vorticity-depletion effects at critical points, which together enable a nontrivial transition threshold in Sobolev perturbations. The results demonstrate rapid metastable relaxation to a near-Kolmogorov shear flow before slow diffusion-dominated decay to zero, and reveal intricate time-scale structure up to . The methods, combining enhanced dissipation, inviscid damping, and vorticity depletion with resolvent and multiscale analysis, provide a flexible blueprint for nonlinear stability problems in high-Reynolds incompressible flows and may extend to other non-monotone shear profiles.

Abstract

In this paper we prove the asymptotic stability of the Kolmogorov flow on a non-square torus for perturbations satisfying , where is the viscosity. Kolmogorov flows are important metastable states to the two dimensional incompressible Navier Stokes equations in the high Reynolds number regime. Our result shows that the perturbed solution will rapidly converge to a shear flow close to the Kolmogorov flow, before settling down to the Kolmogorov flow and slowly decaying to as . In fact, our analysis reveals several interesting time scales and rich dynamical behavior of the perturbation in the transition period . The threshold , which is the same as that for the Couette flow, is quite surprising since one of the key stability mechanisms, enhanced dissipation, becomes considerably weaker in the case of Kolmogorov flows due to the presence of critical points. To overcome this essential new difficulty, we establish sharp vorticity depletion estimates near the critical points to obtain improved decay rates for the vorticity and velocity fields that are comparable with those for Couette flows, at least for our purposes. We then combine these estimates (enhanced dissipation, inviscid damping and vorticity depletion) with a quasilinear approximation scheme and a multiple-timescale analysis naturally adapted to the dynamics of the perturbation, to obtain the threshold for dynamic stability of Kolmogorov flows. The threshold is expected to be sharp when the perturbation is considered in Sobolev spaces. This appears to be the first result that applies vorticity depletion estimates to improve thresholds for nonlinear asymptotic stability in incompressible fluid equations.
Paper Structure (55 sections, 79 theorems, 968 equations)

This paper contains 55 sections, 79 theorems, 968 equations.

Table of Contents

  1. Introduction and main results
  2. Statement of our main result
  3. Background and related work
  4. Transition threshold problems
  5. Related results and Methodology
  6. Enhanced dissipation, Inviscid damping and Vorticity depletion
  7. Main ideas and outline of the proof
  8. Quasilinear approximation method
  9. Construction of the approximate solution
  10. Sharp estimates for the linearized Euler equation around Kolmogorov flows
  11. Spacetime estimates
  12. The three time scales
  13. The reaction term
  14. Dimension counting, anisotropy and time dependent coefficients
  15. Organization of the paper
  16. ...and 40 more sections

Key Result

Theorem 1.1

There exist $\epsilon,\epsilon_1>0$ such that for all $0 < \nu\leq \epsilon_1$, if the initial vorticity $\Omega_0$ satisfies then the solution $\Omega(t)$ of eq:NS-vorticity is global in time and satisfies the following asymptotic stability estimates: for $t\geq 0$, In the above, $\langle a\rangle:=\sqrt{1+|a|^2}$ is the usual Japanese bracket, and is the projection of $h$ off the zero mode in

Theorems & Definitions (156)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • proof
  • ...and 146 more