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Abstract
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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
satisfies
for some small
independent of the
viscosity coefficient
,
then the solution of the 2D Navier–Stokes equations rapidly
converges to some shear flow which is close to Couette flow for
.
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
and long
time scale
separately; (2) we construct a new multiplier to control the growth caused by echo
cascades.
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Keywords
Navier–Stokes equations, Couette flow, inviscid damping,
enhanced dissipation
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Mathematical Subject Classification
Primary: 35Q30
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Milestones
Received: 8 December 2022
Revised: 13 March 2023
Accepted: 28 March 2023
Published: 2 November 2023
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© 2023 MSP (Mathematical Sciences
Publishers). |
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