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

Abstract

We consider the solution to the two-dimensional Navier–Stokes equations around the Poiseuille flow (y²,0) on T x R with small viscosity v > 0 . Via a hypocoercivity argument, we prove that the -dependent modes of the solution to the linear problem undergo the enhanced dissipation effect with a rate proportional to v^1/2 . Moreover, we study the nonlinear enhanced dissipation effect and we establish a transition threshold of v^2/3+ for initial data in L² . Namely, when the L² norm of the perturbation of the Poiseuille flow is size at most v^2/3+ , its size remains so for all times and the enhanced dissipation persists with a rate proportional to v^1/2 .

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