Enhanced dissipation for the 2D Couette flow in critical space

2019/08/29 by Masmoudi, Nader, Zhao, Weiren · 6 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1908.11035

Abstract

We consider the 2D incompressible Navier-Stokes equations on \mathbbT× R, with initial vorticity that is δ close in H^log_xL²_y to -1(the vorticity of the Couette flow (y,0)). We prove that if δ≪ ν^1/2, where ν denotes the viscosity, then the solution of the Navier-Stokes equation approaches some shear flow which is also close to Couette flow for time t≫ ν^-1/3 by a mixing-enhanced dissipation effect and then converges back to Couette flow when t→ +∞. In particular, we show the nonlinear enhanced dissipation and the inviscid damping results in the almost critical space H^log_xL²_y⊂ L²_x,y.

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