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Observability inequalities for transport equations through Carleman estimates. (arXiv:1807.05005v1 [math.AP])
来源于:arXiv
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. 查看全文>>