Inverse problem for a time-dependent Convection-diffusion equation in admissible geometries

TL;DR

This paper addresses a novel inverse problem for a time-dependent convection-diffusion equation on admissible manifolds, demonstrating unique recovery of time-dependent convection and density terms from partial data, a first in Riemannian geometry context.

Contribution

It introduces the first study of partial data inverse problems for recovering time-dependent perturbations in evolution equations on Riemannian manifolds.

Findings

Unique recovery of convection and density terms modulo gauge invariance

First analysis of partial data inverse problem for time-dependent convection-diffusion in Riemannian geometry

Extension of inverse problem theory to time-dependent operators on manifolds

Abstract

We consider a partial data inverse problem for a time-dependent convection-diffusion equation on an admissible manifold. We prove that the time-dependent convection term and time-dependent density can be recovered uniquely modulo a known gauge invariance. There have been several works on inverse problems related to the steady state convection-diffusion operator in Euclidean as well as in Riemannian geometry settings; however, inverse problems related to time-dependent convection-diffusion equation on a manifold are not studied in the prior works, which is the main aim of this paper. In fact, to the best of our knowledge, the problem studied here is the first work related to a partial data inverse problem for recovering both first and zeroth-order time-dependent perturbations of evolution equations in the Riemannian geometry setting.