Mixing and enhanced dissipation in a time-translating shear flow
Motivated in part by the work of Vanneste and Byatt-Smith, we study mixing and enhanced dissipation for the advection-diffusion equation with velocity field $\mathbf{u}(x,y,t)=(\sin(y-ct),0)$, a shear flow whose profile translates rigidly with speed $c$. This is a prototypical example of a flow whose critical points move in time. We quantify how the decay properties depend on the relation between translation speed $c$ and diffusivity $ν$. We first analyse the inviscid transport problem and establish time-averaged $H^{-1}$ mixing estimates for $t\lesssim c^{-1}$, yielding decay rates faster than stationary estimates. Building on these estimates, we prove enhanced dissipation for moderate translation speeds $c=c_0ν^\ell$ with $\ell\in(1/3,3/4)$. In this regime we obtain decay at rate $ν^{(1+2\ell)/5}$, which interpolates continuously between the sharp rates $ν^{1/2}$ for stationary shear flows with simple critical points and $ν^{1/3}$ for monotone flows. This quantifies how increasing translation speed progressively weakens the influence of the critical points. Comparing the inviscid mixing and enhanced dissipation timescales heuristically explains the lower endpoint $\ell=1/3$. For $c\gg 1$, we show that solutions remain close to those of the heat equation on fixed time intervals, such that the rapid translation averages out advection and weakens mixing. The mixing estimate relies on a refined stationary phase analysis exploiting cancellations generated by the motion of the critical points. The enhanced dissipation result requires an adaptation of the hypocoercivity framework for stationary shear flows to the non-autonomous setting. The translating flow prevents the commutator hierarchy from closing in the standard way, which we overcome by constructing an extended energy functional. The large-$c$ analysis exploits the averaging effect of rapid translations in this regime.