The Sobolev stability threshold of 2D hyperviscosity equations for shear flows near Couette flow

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.