Accelerated relaxation enhancing flows cause total dissipation

We show that by "accelerating" relaxation enhancing flows, one can construct a flow that is smooth on $[0,1) \times \mathbb{T}^d$ but highly singular at $t=1$ so that for any positive diffusivity, the advection-diffusion equation associated to the accelerated flow totally dissipates solutions, taking arbitrary initial data to the constant function at $t=1$.