For advection-diffusion on T^d there exist incompressible velocity fields and initial data whose L2 mass decays double exponentially in 2D, like e^{-Ct^2} in 3D, and superexponentially in 4D.
A Stochastic RAGE Theorem and Enhanced Dissipation for Transport Noise
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abstract
We prove a stochastic version of the classical RAGE theorem that applies to the two-point motion generated by noisy transport equations. As a consequence, we identify a necessary and sufficient condition for the corresponding diffusive equation to be dissipation enhancing. This involves the identification of a non-trivial, finite dimensional subspace that is invariant for the family of self-adjoint operator characterizing the structure of the transport noise. We discuss several examples and prove a sharp enhanced dissipation rate for stochastic shear flows.
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Superexponential dissipation enhancement on $\mathbb{T}^d$
For advection-diffusion on T^d there exist incompressible velocity fields and initial data whose L2 mass decays double exponentially in 2D, like e^{-Ct^2} in 3D, and superexponentially in 4D.