TY - JOUR

T1 - The Sobolev Stability Threshold for 2D Shear Flows Near Couette

AU - Bedrossian, Jacob

AU - Vicol, Vlad

AU - Wang, Fei

N1 - Funding Information:
Acknowledgements The work of J.B. was partially supported by NSF Grant DMS-1462029, and by an Alfred P. Sloan Research Fellowship. The work of V.V. was partially supported by NSF Grant DMS-1514771 and by an Alfred P. Sloan Research Fellowship. The work of F.W. was partially supported by NSF Grant DMS-1514771.
Funding Information:
The work of J.B. was partially supported by NSF Grant DMS-1462029, and by an Alfred P. Sloan Research Fellowship. The work of V.V. was partially supported by NSF Grant DMS-1514771 and by an Alfred P. Sloan Research Fellowship. The work of F.W. was partially supported by NSF Grant DMS-1514771.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - We consider the 2D Navier–Stokes equation on T× R, with initial datum that is ε-close in HN to a shear flow (U(y), 0), where ‖U(y)-y‖HN+4≪1 and N> 1. We prove that if ε≪ ν1 / 2, where ν denotes the inverse Reynolds number, then the solution of the Navier–Stokes equation remains ε-close in H1 to (etν∂yyU(y),0) for all t> 0. Moreover, the solution converges to a decaying shear flow for times t≫ ν- 1 / 3 by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than ν1 / 2 for 2D shear flows close to the Couette flow.

AB - We consider the 2D Navier–Stokes equation on T× R, with initial datum that is ε-close in HN to a shear flow (U(y), 0), where ‖U(y)-y‖HN+4≪1 and N> 1. We prove that if ε≪ ν1 / 2, where ν denotes the inverse Reynolds number, then the solution of the Navier–Stokes equation remains ε-close in H1 to (etν∂yyU(y),0) for all t> 0. Moreover, the solution converges to a decaying shear flow for times t≫ ν- 1 / 3 by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than ν1 / 2 for 2D shear flows close to the Couette flow.

KW - Enhanced dissipation

KW - Inviscid damping

KW - Stability of shear flows

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U2 - 10.1007/s00332-016-9330-9

DO - 10.1007/s00332-016-9330-9

M3 - Article

AN - SCOPUS:84983469021

VL - 28

SP - 2051

EP - 2075

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 6

ER -