# Inverse coefficient problems for a transport equation by local Carleman estimate

## TL;DR

This paper addresses inverse problems for a transport equation, establishing conditional stability results for determining coefficients using Carleman estimates that depend on the unknown functions.

## Contribution

It introduces a novel Carleman estimate with a weight function depending on the unknown coefficient, enabling stability analysis for inverse coefficient problems.

## Key findings

- Conditional Hölder stability in a subdomain for coefficient determination.
- Stability depends on the outward normal component of H on the boundary.
- The method applies Carleman estimates tailored to the transport equation.

## Abstract

We consider the transport equation $\ppp_tu(x,t) + (H(x)\cdot \nabla u(x,t)) + p(x)u(x,t) = 0$ in $\OOO \times (0,T)$ where $\OOO \subset \R^n$ is a bounded domain, and discuss two inverse problems which consist of determining a vector-valued function $H(x)$ or a real-valued function $p(x)$ by initial values and data on a subboundary of $\OOO$. Our results are conditional stability of H\"older type in a subdomain $D$ provided that the outward normal component of $H(x)$ is positive on $\ppp D \cap \ppp\OOO$. The proofs are based on a Carleman estimate where the weight function depends on $H$.