We consider the transport equation ∂tu(x, t) + H(t) · ∇ u(x, t) = 0 in Ω× (0, T), where T> 0 and Ω⊂ Rd is a bounded domain with smooth boundary ∂Ω. 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 ∂Ω, we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.
Observability Inequalities for Transport Equations through Carleman Estimates / Cannarsa, P.; Floridia, G.; Yamamoto, M.. - 32:(2019), pp. 69-87. [10.1007/978-3-030-17949-6_4]
Observability Inequalities for Transport Equations through Carleman Estimates
Cannarsa P.;Floridia G.
;Yamamoto M.
2019
Abstract
We consider the transport equation ∂tu(x, t) + H(t) · ∇ u(x, t) = 0 in Ω× (0, T), where T> 0 and Ω⊂ Rd is a bounded domain with smooth boundary ∂Ω. 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 ∂Ω, we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
