Poon-type logarithmic lower bound for Sobolev advection diffusion

Fix $p\in[1,+\infty)$. Let $b\in C^{\infty}([0,T]\times\mathbb{T}^d)$ be divergence free and let $u_0\in W^{1,2}(\mathbb{T}^d)$. If $u^\nu$ is a solution of the advection diffusion equation, then Poon-type logarithmic lower-bound conjecture. For every $\nu\in(0,1)$ and $t>0$,

$$\left\lVert u_t\right\rVert_{L^2}^2\geq \left\lVert u_0\right\rVert_{L^2}^2\exp\left\lbrace-\log(1/\nu)^{-p}C_1\int_0^t\exp\left\lbrace C_2\int_0^s\left\lVert\nabla b_r\right\rVert_{L^p}\mathop{}\!\mathrm{d}r\right\rbrace\mathop{}\!\mathrm{d}s\right\rbrace,$$

where $C_1=C_1(u_0,p,d)>0$ and $C_2=C_2(p,d)>0$. This is proposed as a natural Sobolev analogue of Poon's smooth-velocity estimate; the paper presents it as a conjectural lower bound.