3 research outputs found
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 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 which guarantees that the orbits of intersect
\ppp\OOO, we prove an energy estimate which in turn yields an observability
inequality. Our results are motivated by applications to inverse problems.Comment: 18 pages, 3 figure