Inverse boundary value problem for the Convection-Diffusion equation with local data

TL;DR

This paper investigates the inverse problem of determining time-dependent convection and density terms in a Convection-Diffusion Equation using boundary data, addressing challenges posed by inaccessible boundary parts and gauge invariances.

Contribution

It establishes uniqueness of the inverse problem under certain geometric conditions and characterizes the gauge invariance affecting the data interpretation.

Findings

Proves unique determination of convection and density terms from boundary data.

Identifies the gauge invariance as the only obstruction to uniqueness.

Handles the case with an inaccessible boundary part under flatness assumption.

Abstract

We study a local data inverse problem for the time-dependent Convection-Diffusion Equation (CDE) in a bounded domain where a part of the boundary is treated to be inaccessible. Up on assuming the inaccessible part to be flat, we seek for the unique determination of the time-dependent convection and the density terms from the knowledge of the boundary data measured outside the inaccessible part. In the process, we show that there is a natural gauge in the perturbations, and we prove that this is the only obstruction in the uniqueness result.