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Erschienen in:
2019 | OriginalPaper | Buchkapitel
Observability Inequalities for Transport Equations through Carleman Estimates
verfasst von : Piermarco Cannarsa, Giuseppe Floridia, Masahiro Yamamoto
Erschienen in: Trends in Control Theory and Partial Differential Equations
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Abstract
We consider the transport equation \(\partial _t u(x,t) + H(t)\cdot \nabla u(x,t) = 0\) in \(\varOmega \times (0,T),\) where \(T>0\) and \(\varOmega \subset \mathbb R^d \) is a bounded domain with smooth boundary \(\partial \varOmega \). 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 \(\partial \varOmega \), we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.