A Universal Total Anomalous Dissipator

Abstract For all $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) , we construct an explicit divergence-free vector field $$V\in L^\infty _tC^\alpha _x \cap C^{\frac{\alpha }{1-\alpha }}_t L^\infty _x$$ V ∈ L t ∞ C x α ∩ C t α 1 - α L x ∞ so that the solutions to the drift–diffusion equations $$\begin{aligned} \partial _t\theta ^\kappa -\kappa \Delta \theta ^\kappa +V\cdot \nabla \theta ^\kappa =0 \end{aligned}$$ ∂ t θ κ - κ Δ θ κ + V · ∇ θ κ = 0 exhibit asymptotic total dissipation for all mean-zero initial data: $$\lim _{\kappa \rightarrow 0}\Vert \theta ^\kappa (1,\cdot )\Vert _{L^2}=0$$ lim κ → 0 ‖ θ κ ( 1 , · ) ‖ L 2 = 0 . Additionally, we give explicit rates in $$\kappa $$ κ and uniform dependence on initial data.