Stably determining time-dependent convection-diffusion coefficients from a partial Dirichlet-to-Neumann map

We study in this paper the inverse problem for the dynamical convection-diffusion equation. More precisely, we set logarithmic stability estimates in the determination of the two time-dependent first-order convection term and the scalar potential appearing in the heat equation. The observations here are taken only on an arbitrary open subset of the boundary and are given by a partial Dirichlet-to-Neumann map. For this end, we will reduce our initial problem into an auxiliary one then we will construct particular solutions and apply a special parabolic Carleman estimate.