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An algebraic lower bound on the energy decay for solutions of theadvection-diffusion equation in $\mathbb{R}^d$ with $d=2,3$ is derived usingthe Fourier splitting method. Motivated by a conjecture on mixing of passivescalars in fluids, a lower bound on the $L^2-$ norm of the inverse gradient ofthe solution is obtained via gradient estimates and interpolation.

Lower bounds on mixing norms for the advection diffusion equation in $\mathbb{R}^d$