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The Sobolev stability threshold of 2D hyperviscosity equations for shear flows near Couette flow
Author(s) -
Luo Xiang
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6372
Subject(s) - hyperviscosity , couette flow , mathematics , shear flow , flow (mathematics) , shear (geology) , mathematical analysis , viscosity , mechanics , physics , thermodynamics , geometry , geology , blood viscosity , medicine , petrology , cardiology
We consider the 2D hyperviscosity equations on T × R . We show that if the initial data of 2D hyperviscosity equations are ϵ ‐close to the shear flows ( U ( y ),0), which are sufficiently small perturbations of Couette flow ( y ,0), then the solution will stay ϵ ‐close to ( e − ν t ∂ y 4U ( y ) , 0 ) for all t >0, where ϵ ≪ ν1 2and ν denotes the kinematic viscosity coefficient. What is more, by the mixing‐enhanced effect, the solutions converge to decaying shear flows for t ≫ ν − 1 5, which is faster than the heat‐equation timescale. Hence, we conclude that the stability threshold of 2D hyperviscosity equations with initial data ( U ( y ),0) is not worse thanν1 2.