Inverse problem for a transport equation using Carleman estimates
Résumé
This article is devoted to prove a stability result for an absorption coefficient for a transport equation in a smooth domain $\Omega$ with $\partial \Omega=\Gamma^+ \cup \Gamma^-$. We consider the transport equation $ \partial_t u +\nabla a \cdot \nabla u + V(x)u=0,\ \ \mbox{ in } \ \Omega \times (0,T)$ with boundary condition $u(x,t)=h(x,t),\ \ \mbox{ on } \ \Gamma^- \times (0,T)$ and initial condition $u(x,t)=u_0(x),\ \ x \in \Omega$, where $a$ is such that $\nabla a \cdot \nu >0$ on $\Gamma^+$ and $\nabla a \cdot \nu <0$ on $\Gamma^-$. The result is obtained using a global Carleman estimate with only one observation on a part $\Gamma^+$ of the boundary and the data of the solution at a fixed time on the whole domain.
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