Determining time-dependent convection and density terms in a convection-diffusion equation using partial data

This article studies an inverse boundary value problem for the time-dependent convection-diffusion equation. We use the nonlinear Carleman weight to uniquely recover the time-dependent convection term and time-dependent density coefficient. Nonlinear weight allows us to prove the uniqueness of the coefficients by making measurements on possibly a small subset of the boundary. We show that the convection term and the density coefficient can be recovered up to the natural gauge from the knowledge of the Dirichlet to Neumann map measured on a small open subset of the boundary.