*preprint*

**Inserted:** 27 jul 2018

**Year:** 2018

**Abstract:**

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 $\ppp\OOO$. First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition on $H$ which guarantees that the orbit $\{ H(t)\in\R^d, \thinspace 0 \le t \le T\}$intersects $\ppp\OOO$, we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.