Enhanced Dissipation and Transition Threshold for the Poiseuille Flow in a Periodic Strip

We consider the solution to the 2D Navier-Stokes equations around the Poiseuille flow (y, 0) on T × R with small viscosity ν > 0. Via a hypocoercivity argument, we prove that the x−dependent modes of the solution to the linear problem undergo the enhanced dissipation effect with a rate proportional to ν 1 2 . Moreover, we study the nonlinear enhanced dissipation effect and we establish a transition threshold of ν 2 3 . Namely, when the perturbation of the Poiseuille flow is size at most ν 2 3 , its size remains so for all times and the enhanced dissipation persists with a rate proportional to ν 1 2 .

[1]  N. Masmoudi,et al.  Stability threshold of two-dimensional Couette flow in Sobolev spaces , 2019, Annales de l'Institut Henri Poincaré C, Analyse non linéaire.

[2]  S. Ibrahim,et al.  On Pseudospectral Bound for Non-selfadjoint Operators and Its Application to Stability of Kolmogorov Flows , 2017, Annals of PDE.

[3]  Michele Coti Zelati,et al.  On the Relation between Enhanced Dissipation Timescales and Mixing Rates , 2018, Communications on Pure and Applied Mathematics.

[4]  L. Rayleigh.  On the Stability, or Instability, of certain Fluid Motions , 1879 .

[5]  Nader Masmoudi,et al.  Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case , 2015, Memoirs of the American Mathematical Society.

[6]  Dongyi Wei.  Diffusion and mixing in fluid flow via the resolvent estimate , 2018, Science China Mathematics.

[7]  Zhifei Zhang,et al.  Transition Threshold for the 3D Couette Flow in Sobolev Space , 2018, Communications on Pure and Applied Mathematics.

[8]  Zhifei Zhang,et al.  Enhanced dissipation for the Kolmogorov flow via the hypocoercivity method , 2019, Science China Mathematics.

[9]  Vlad Vicol,et al.  Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow , 2014, 1408.4754.

[10]  P. Constantin,et al.  Diffusion and mixing in fluid flow , 2005 .

[11]  S. Ding,et al.  Enhanced dissipation and transition threshold for the 2-D plane Poiseuille flow via resolvent estimate , 2020, Journal of Differential Equations.

[12]  O. Reynolds.  III. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels , 1883, Proceedings of the Royal Society of London.

[13]  Fei Wang,et al.  The Sobolev Stability Threshold for 2D Shear Flows Near Couette , 2016, J. Nonlinear Sci..

[14]  Nader Masmoudi,et al.  On the stability threshold for the 3D Couette flow in Sobolev regularity , 2015, 1511.01373.

[15]  Michele Coti Zelati,et al.  Enhanced Dissipation in the Navier–Stokes Equations Near the Poiseuille Flow , 2019, 1901.01571.

[16]  Zhifei Zhang,et al.  Linear Stability of Pipe Poiseuille Flow at High Reynolds Number Regime , 2019, Communications on Pure and Applied Mathematics.

[17]  Zhifei Zhang,et al.  Linear inviscid damping and enhanced dissipation for the Kolmogorov flow , 2017, Advances in Mathematics.