Observability inequalities for transport equations through Carleman estimates

We consider the transport equation $\ppp_t u(x,t) + %\alpha' H(t)\cdot \nabla u(x,t) = 0$ in $\OOO\times(0,T),$ where $T>0$ and $\OOO\subset \R^d%,\, d\in\N, $ is a bounded domain with smooth boun\-dary $\ppp\OOO$. First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition which guarantees that the orbits of $H$ intersect $\ppp\OOO$, we prove an energy estimate which in turn yields an obser\-vability inequality. Our results are motivated by applications to inverse problems.