This thesis provides a quantitative analysis of scalar decay for the advection-diffusion equation on the two-dimensional torus with time-dependent shear flows. Advection generates fine-scale structures, a process called mixing, on which diffusion acts efficiently. This accelerates decay beyond the purely diffusive rate, a phenomenon known as enhanced dissipation. For time-independent flows, both mixing and enhanced dissipation are governed by the critical points of the velocity profile. This work extends the theory to non-autonomous settings, studying how time-dependence affects decay rates.

Two flow classes are studied. The first is a product flow with fixed spatial profile and time-varying amplitude. We prove that both mixing and enhanced dissipation are governed by an integral of the amplitude modulation. This can produce decay faster than any stationary flow with the same profile, or permit the velocity to vanish at certain times while retaining decay. The second class is a translating sine flow, for which we establish a fast mixing estimate in a time-averaged sense. For enhanced dissipation, at intermediate translation speeds the rate interpolates between those for simple critical points and monotone profiles, accelerating decay over the stationary case. At large translation speeds, advection averages out any enhancing effect and the decay approaches the diffusive timescale. We provide numerical computations supporting these rates.

The proofs of these results also require methodological extensions. For the enhanced dissipation estimates, we employ the hypocoercivity framework, departing from the constructions for stationary flows. Product flows require time-dependent weights that dynamically rebalance the energy functional. For the translating flow, the commutator hierarchy does not close at finite order, and we construct an extended functional to track the transfer of energy across oscillating modes. The mixing estimate combines a time-averaged formulation with a stationary phase argument in space and time to capture the cancellations generated by the moving critical points.