Enhanced dissipation for the 2D couette flow in critical space. Issue 12 (20th July 2020)

• Record Type: Journal Article
• Authors: Masmoudi, Nader; Zhao, Weiren
• Is Part Of: Communications in partial differential equations. Volume 45:Issue 12(2020)
• Page Start: 1682
• Page End: 1701
• Publication Date: 2020-07-20
• Subjects: Couette flow; critical space; enhanced dissipation; inviscid damping, Navier Stokes; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2020.1791180
• Languages: English
• ISSNs: 0360-5302

Abstract

Abstract: We consider the 2 D incompressible Navier-Stokes equations on T × R, with initial vorticity that is δ close in H x log L y 2 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 x log L y 2 ⊂ L x, y 2 .