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【数学系邀请报告】Transition threshold problem for the 3-D Couette flow in a finite channel

Abstract: Understanding the transition mechanism from the laminar flow to the turbulent flow is a longstanding problem in fluid mechanics.  For this, Trefethen et al(Science 1993) proposed to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. In this talk, I will introduce a joint work on the transition threshold for 3-D Couette flow(with Qi Chen and Dongyi Wei). We showed that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}$ for some $c_0>0$ independent of $Re$, then the solution of the 3-D Navier-Stokes equations is global in time and does not transition away from the Couette flow in the $L^\infty$ sense, and rapidly converges to a streak solution for $t\gtrsim Re^{1/3}$ due to the mixing-enhanced dissipation effect. This result also confirms the transition threshold conjecture proposed by Trefethen et al(Science 1993).