Enhanced dissipation and transition threshold for the 2-D plane Poiseuille flow via resolvent estimate

2020/08/23 by Ding, Shijin, Lin, Zhilin · 4 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2008.10057

Abstract

In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Poiseuille flow (1-y²,0) in a finite channel with Navier-slip boundary condition. Based on the resolvent estimates for the linearized operator around the Poiseuille flow, we first establish the enhanced dissipation estimates for the linearized Navier-Stokes equations with a sharp decay rate e^-c√νt. As an application, we prove that if the initial perturbation of vortiticy satisfies ‖ω₀‖_L²≤ c₀ν^(3)/(4), for some small constant c₀>0 independent of the viscosity ν, then the solution dose not transition away from the Poiseuille flow for any time.

Cited by

• Transition threshold for the Navier-Stokes-Coriolis system at high Reynolds numbers
• Quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition
• The optimal transition threshold for the 2D Couette flow in the infinite channel
• Enhanced dissipation and Taylor dispersion by a parallel shear flow in an infinite cylinder with unbounded cross section

Related

• Transition threshold for the 2-D Couette flow in a finite channel
• Enhanced Dissipation and Transition Threshold for the Poiseuille Flow in a Periodic Strip
• Stability for the 2-D plane Poiseuille flow in finite channel
• Transition threshold for the 2-D Couette flow in whole space via Green's function
• The optimal transition threshold for the 2D Couette flow in the infinite channel