Inverse coefficient problems for a transport equation by local Carleman estimate

We consider the transport equation $\ppp_tu(x,t) + (H(x)\cdot \nabla u(x,t))

p(x)u(x,t) = 0 $i n$ \OOO \times (0,T) $w h e r e$ \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 $H$ .

We consider the transport equation $\ppp_tu(x,t) + (H(x)\cdot \nabla u(x,t))

p(x)u(x,t) = 0 $i n$ \OOO \times (0,T) $w h e r e$ \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 $H$ .

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