Exponential Lower Bounds for the Advection-Diffusion Equation with Shear Flows

In this paper, we prove that the $L^2$ norm of spatial mean-free solutions to the advection–diffusion equation on $\mathbb{T}^2$ with shear drifts satisfies an \emph{exponential lower bound} in time. This lower bound shows that diffusion can fundamentally suppress passive-scalar mixing.

💡 Research Summary

The paper investigates the long‑time behavior of solutions to the advection–diffusion equation on the two‑dimensional torus 𝕋² with a shear drift. The equation under study is

 ∂ₜρ + U(t,y)∂ₓρ = μΔρ, ρ(0,·)=ρ₀, ∫_{𝕋²}ρ₀=0,

where U(t,y) is a divergence‑free, L²‑bounded shear flow, μ>0 is the diffusion coefficient, and the initial data has zero spatial mean. The authors focus on the interplay between filamentation (stretching and folding caused by advection) and homogenization (diffusive smoothing). While many works have shown that filamentation can accelerate homogenization—so‑called enhanced dissipation—this paper addresses the opposite direction: can diffusion suppress filamentation and thereby limit mixing?

The main result (Theorem 2.2) establishes that there exist positive constants c₁<c₂, depending only on U and ρ₀, such that for all t≥0

 c₁ e^{‑c₂t} ≤ ‖ρ(t)‖_{L²} ≤ c₂ e^{‑c₁t}.

Thus the L²‑norm cannot decay faster than a simple exponential; a double‑exponential lower bound is impossible. The proof proceeds by contradiction: assuming a faster-than‑exponential decay, the authors expand the solution in Fourier series in the x‑direction, ρ(t,x,y)=∑_{k∈ℤ}ρ_k(t,y)e^{ikx}, and derive an evolution equation for each mode

 ∂ₜρ_k + (k²‑∂_{yy})ρ_k = –ik U(t,y) ρ_k.

Introducing a time‑dependent exponential weight Λ_m = k² + m² + (m+1)²/2 and defining h(t,y)=e^{Λ_m t}ρ_k(t,y), they transform the problem into a forced linear equation with constant coefficients in the Fourier‑frequency domain. By taking the temporal Fourier transform, they obtain an explicit representation

 ĥ_l(τ) = (iτ + k² + l² – Λ_m)^{-1}