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

.We consider the solution to the two-dimensional Navier–Stokes equations around the Poiseuille flow \((y^2,0)\) on \(\mathbb{T}\times \mathbb{R}\) with small viscosity \(\nu \gt 0\) . 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 \(\nu^{\frac{1}{2}}\) . Moreover, we study the nonlinear enhanced dissipation effect and we establish a transition threshold of \(\nu^{\frac{2}{3}+}\) for initial data in \(L^2\) . Namely, when the \(L^2\) norm of the perturbation of the Poiseuille flow is size at most \(\nu^{\frac{2}{3}+}\) , its size remains so for all times and the enhanced dissipation persists with a rate proportional to \(\nu^{\frac{1}{2}}\) .KeywordsNavier–Stokes equationsPoiseuille flowtransition thresholdenhanced dissipationmetastabilityMSC codes35Q3035Q3576D0576E05