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.

Automated summary

In this paper, we investigate the inverse problem for the dynamical convection-diffusion equation, setting logarithmic stability estimates in determining the time-dependent convection term and scalar potential. Observations are made on an arbitrary open subset of the boundary and given by a partial Dirichlet-to-Neumann map. The initial problem is reduced to an auxiliary one, with particular solutions constructed and a special parabolic Carleman estimate applied.