Bressan's conjecture on the geometric mixing scale

Let $\theta$ be a passive scalar transported on $\mathbf{R}^d$ by a time-dependent divergence-free velocity field $u$, and let $\theta(t,\cdot)$ denote the resulting scalar. Write $\varepsilon(t)$ for the geometric mixing scale of $\theta(t,\cdot)$, and write

$$\lVert \nabla u(t',\cdot) \rVert_1$$

for the $L^1$ norm of the spatial gradient of the velocity. Bressan's conjecture. There exists a constant $C>0$ depending on $\theta_0$ such that

$$\varepsilon(t)\ge C^{-1}\exp\left(-C\int_0^t\lVert \nabla u(t',\cdot) \rVert_1\,dt'\right).$$

The conjecture predicts an exponential lower bound on the geometric mixing scale in terms of the accumulated $L^1$ norm of the velocity gradient. Existing results had established analogous bounds under an $L^p$ assumption with $p>1$, while this $L^1$ formulation is the conjectural endpoint addressed by the paper's approach.