Transition Threshold for the3DCouette Flow in Sobolev Space

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds numberRe. It was proved that if the initial velocityv(0)satisfies parallel to v0-(y,0,0)parallel to H2 <= c0Re-1for somec(0) > 0independent ofRe, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture proposed by Trefethen et al. in 1993. Moreover, we prove that the long-time dynamics of the solution behaves as: fort less than or similar to Re, the solution will experience a transient growth fromO(Re-1)toO(c(0))due to the 3D lift-up effect; fort greater than or similar to Re-1/3, the solution will rapidly converge to a streak solution due to the mixing-enhanced dissipation effect. (c) 2020 Wiley Periodicals LLC

Automated summary

This paper studies the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number Re. It was proved that under certain conditions, the solution of the 3D Navier-Stokes equations will remain near the Couette flow. The long-time dynamics of the solution can be divided into transient growth and mixing-enhanced dissipation.