Observability Inequalities for Transport Equations through Carleman Estimates

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.

Appendix

In this appendix we prove Lemma 1.2.

Proof

(of Lemma 1.2). Since $H\in Lip([0,T];\mathbb R^d)$ there exists $L>0$ such that

$$\vert H(t)-H(s)\vert \le L|t-s|,\;\forall t,s\in [0,T].$$

Let us consider, for simplicity, a uniform partition $\{t_j\}_0^m$ of [0, T]. Let us set

$$ \displaystyle \eta _j:=\frac{H(t_j)}{|H(t_j)|},\;j=0\ldots ,m-1. $$

For $t \in [t_j,t_{j+1}]$, $j=0, ..., m-1$, we deduce

$$\begin{aligned} H(t) \cdot \eta _j= & {} \left( H(t)-H(t_j)\right) \cdot \eta _j+H(t_j)\cdot \eta _j\ge -|H(t)-H(t_j)|+|H(t_j)| \nonumber \\\ge & {} -L|t-t_j|+|H(t_j)|\ge -L\frac{T}{m}+|H(t_j)|, \end{aligned}$$

(48)

and, since $|H(t)|\le \left| H(t)-H(t_j)\right| +|H(t_j)|,$

$$\begin{aligned} \left| H(t_j)\right| \ge |H(t)|-|H(t)-H(t_j)|\ge |H(t)|-L|t-t_j| \ge |H(t)|-L\frac{T}{m}. \end{aligned}$$

(49)

From (48) and (49), if we choose the uniform partition with $m\ge \frac{2 LT}{H_0(1-S_*)}$, where we recall that $\displaystyle H_0=\min _{t\in [0,T]}|H(t)|$, we obtain the conclusion, that is,

$$ H(t)\cdot \frac{H(t_j)}{|H(t_j)|}\ge |H(t)|-2L\frac{T}{m}\ge S_* |H(t)|, \;\;\;\forall t\in [t_{j},t_{j+1}], \;\;\forall j=0,\ldots ,m-1. $$

   $\square$