Observability inequalities for transport equations through Carleman estimates

TL;DR

This paper establishes observability inequalities for transport equations using Carleman estimates, providing tools for inverse problems by linking boundary measurements to internal states under specific conditions.

Contribution

It introduces new Carleman estimates for transport equations with piecewise continuous weights and derives observability inequalities under orbit intersection conditions.

Findings

Carleman estimate for finite energy solutions

Observability inequality under orbit intersection condition

Applications to inverse problems

Abstract

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 which guarantees that the orbits of $H$ intersect $\ppp\OOO$, we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.