Bressan's conjecture for anomalous dissipation with rough velocity fields

Let $\theta^\kappa$ be a solution of the viscous transport equation on $\mathbb{T}^d$ with velocity field $u\in L^1([0,\infty),W^{1,1}(\mathbb{T}^d))$ and smooth initial data $\theta_0$. Let $r(\kappa)$ be a universal rate function, independent of $u$, with $r(\kappa)\to0$ as $\kappa\to0$. Bressan's dissipation conjecture. For every $0\leqslant\kappa\leqslant1$,

$$|\theta^\kappa|_{L^2}\geqslant |\theta_0|_{L^2}\exp\left(-r(\kappa)C_1(\theta_0)\int_0^t\exp\left(C_2\int_0^s|\nabla u(\cdot,s')|_{L^1}\,\mathrm{d}s'\right)\,\mathrm{d}s\right),$$

where $C_1>0$ depends on $\theta_0$ but not on $\kappa$, and $C_2>0$ is universal. In particular,

$$\kappa\int_0^t|\nabla\theta^\kappa|_{L^2}^2\,\mathrm{d}s\leqslant |\theta_0|_{L^2}^2\left(1-\exp\left(-2r(\kappa)C_1(\theta_0)\int_0^t\exp\left(C_2\int_0^s|\nabla u(\cdot,s')|_{L^1}\,\mathrm{d}s'\right)\,\mathrm{d}s\right)\right),$$

and hence, for $\kappa\ll1$, $\kappa\int_0^t|\nabla\theta^\kappa|_{L^2}^2\,\mathrm{d}s\lesssim r(\kappa)$. The conjecture proposes a diffusion-dependent dissipation rate that tends to zero uniformly in the velocity field as $\kappa\to0$. Its status is not resolved by the supplied text.