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Nonlinear enhanced dissipation and inviscid damping for the 2D Couette flow

Dongyi Wei and Zhifei Zhang

Vol. 5 (2023), No. 3, 573–592
Abstract

We study the stability of the Couette flow under the 2D Navier–Stokes equations with full dissipation or only vertical dissipation. It was proved that if the initial velocity v0 satisfies v0 (y,0)H3 cν13 for some small c independent of the viscosity coefficient ν, then the solution of the 2D Navier–Stokes equations rapidly converges to some shear flow which is close to Couette flow for t ν13. Moreover, we prove the optimal enhanced dissipation estimate of the vorticity and inviscid damping estimate of the velocity. To this end, we introduce two new ideas: (1) we make the energy estimates in short time scale t ν16 and long time scale t ν16 separately; (2) we construct a new multiplier to control the growth caused by echo cascades.

Keywords
Navier–Stokes equations, Couette flow, inviscid damping, enhanced dissipation
Mathematical Subject Classification
Primary: 35Q30
Milestones
Received: 8 December 2022
Revised: 13 March 2023
Accepted: 28 March 2023
Published: 2 November 2023
Authors
Dongyi Wei
School of Mathematical Sciences
Peking University
Beijing
China
Zhifei Zhang
School of Mathematical Sciences
Peking University
Beijing
China