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.
Publisher Copyright:
© 2016, Springer Science+Business Media New York.
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
SN - 0938-8974
VL - 28
SP - 2051
EP - 2075
JO - Journal of Nonlinear Science
JF - Journal of Nonlinear Science
IS - 6
ER -