Regularity properties of passive scalars with rough divergence-free drifts

Abstract

We present sharp conditions on divergence-free drifts in Lebesgue spaces for the passive scalar advection-diffusion equation \[ \partial_t θ- Δθ+ b \cdot \nabla θ= 0 \] to satisfy local boundedness, a single-scale Harnack inequality, and upper bounds on fundamental solutions. We demonstrate these properties for drifts $b$ belonging to $L_t^q{L}_x^p$, where $\frac{2}{q}+\frac{n}{p}<2$, or $L_x^p{L}_t^q$, where $\frac{3}{q}+\frac{n-1}{p}<2$. For steady drifts, the condition reduces to $b\in L^{\frac{n-1}{2}+}$. The space $L_t^1{L}_x^{\infty }$ of drifts with `bounded total speed' is a borderline case and plays a special role in the theory. To demonstrate sharpness, we construct counterexamples whose goal is to transport anomalous singularities into the domain `before' they can be dissipated.

33 pages, 3 figures

Cited by in corpus (1)

• A trajectorial interpretation of Moser's proof of the Harnack inequality