2 problems

Conjecture on the diffusion restriction. The condition $\alpha<\nu$ is not necessary for the approximate controllability result: the restriction $\alpha<\nu$ is purely technical.

Uniqueness conjecture. If $abla bar{theta}in L^dx$ and $bar{u}in dot{B}{p,1}^{-1+frac{d}{p}}$, then uniqueness should be provable with the velocity field belonging to