Divergence-free drifts decrease concentration

We show that bounded divergence-free vector fields $u:[0,\mathrm{\infty })\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ decrease the ‘‘concentration’’, quantified by the modulus of absolute continuity with respect to the Lebesgue measure, of solutions to the associated advection-diffusion equation when compared to solutions to the heat equation. In particular, for symmetric decreasing initial data, the solution to the advection-diffusion equation has (without a prefactor constant) larger variance, larger entropy, and smaller $L^p$ norms for all $p\in [1,\mathrm{\infty }]$ than the solution to the heat equation. We also note that the same is not true on ${\mathbb{T}}^d$.