On uniqueness for a rough transport-diffusion equation (Q2630179)

A linear transport-diffusion equation \(\partial_tu+\nabla\cdot(av)=\Delta a\) is studied with a divergence free vector field \(v\) of low regularity \(v\in L^2({\mathbb R}^+,\dot H^1({\mathbb R}^3))\). The main result is uniqueness of solutions to the Cauchy problem. Its proof is based on a duality argument.