Divergence-free drifts decrease concentration

We show that bounded divergence-free vector fields u:[0,∞)×R^d→R^d decrease (that is, do not increase) 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∈[1,∞] than the solution to the heat equation. We also note that the same is not true on T^d[jls-end-space/].

Duke Scholars

Author Renaud Raquépas Mathematics