Observability inequalities for transport equations through Carleman estimates

Abstract

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