中国科学院数学与系统科学研究院
数学研究所
数学科学全国重点实验室
杰出访问研究员报告
Speaker: 郏浩（明尼苏达大学）
Inviter: 付保华
Language: English
Title: Asymptotic metastability of the Kolmogorov flow on a non-square torus
Time & Venue: 2026年9月20日（星期日）10:30-11:30 南楼N913
Abstract: An important phenomenon in the slightly viscous two-dimensional Navier–Stokes equations is the emergence of metastable coherent states from quite general smooth initial data. On a non-square torus, it was observed numerically that generic solutions rapidly relax to shear flows (the so-called bar states) over a timescale much shorter than the diffusion timescale, before slowly converging to the Kolmogorov flow due to viscous decay. Explaining such rapid convergence to metastable states turned out to be quite difficult. Early attempts were made by Beck and Wayne, who studied a model linearized problem and determined the exponent for enhanced dissipation for the model. However, major difficulties remained in extending such an analysis to the full linearized equation or the nonlinear Navier–Stokes equation. In a recent work joint with Qi Chen, Dongyi Wei and Zhifei Zhang, we proved that for vorticity perturbations of the Kolmogorov flow in H³ of size ≪ ν¹ᐟ³ where ν is the small viscosity, the solution rapidly converges to shear flows close to the Kolmogorov flow over the surprisingly short timescale ν⁻¹ᐟ³, in comparison with the viscous timescale ν⁻¹. The threshold ν¹ᐟ³ is conjectured to be sharp for Sobolev perturbations. The proof combines precise estimates for the linearized Navier–Stokes and Euler equations that quantify three essential relaxation mechanisms for incompressible fluid flows: enhanced dissipation, inviscid damping and vorticity depletion. In the talk, we will show some numerical simulations of the dynamics, explain the role of a non-square torus as supposed to the square torus, and briefly go over some key ideas of the proof.
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