Observability inequalities for transport equations through Carleman estimates

We consider the transport equation $\ppp_t u(x,t) + H(t)\cdot \nabla u(x,t) = 0$in$\OOO\times(0,T),$where$T>0$and$\OOO\subset \R^d $ is a bounded domain with smooth boundary $\text{ppp}\text{OOO}$. 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 $\text{ppp}\text{OOO}$, we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.