Carleman estimate with piecewise weight and applications to inverse problems for first-order transport equations

We consider a first-order transport equation $\ppp_tu(x,t) + (H(x)\cdot\nabla u(x,t)) + p(x)u(x,t) = F(x,t)$ for $x \in \OOO \subset \R^d$, where $\OOO$ is a bounded domain and $0<t<T$. We prove a Carleman estimate for more generous condition on the principal coefficients $H(x)$ than in the existing works. The key is the construction of a piecewise smooth weight function in $x$ according to a suitable decomposition of $\OOO$. Our assumptions on $H$ generalize the conditions in the existing articles, and require that a directed graph created by the corresponding stream field has no closed loops. Then, we apply our Carleman estimate to two inverse problems of determinination of an initial value and one of a spatial factor of a source term, so that we establish Lipschitz stability estimates for the inverse problems.