In this paper we prove an inverse inequality for the parabolic equation
$\[u_{t}-\varepsilon\Delta u+M\cdot\nabla u=g\mathbh{1}_{\omega}\]$
in a bounded domain Ω⊂
$\mathbb{R}^{n}$
with Dirichlet boundary conditions. With the motivation of finding an estimate of g in terms on the trace of the solution in 𝒪×(0,T) for ε small, our approach consists in studying the convergence of the solutions of this equation to the solutions of some transport equation when ε→0, and then recover some inverse inequality from the properties of the last one. Under some conditions on the open sets ω, 𝒪 and the time T, we are able to prove that, in the particular case when g∈H01(ω) and it does not depend on time, we have: |g|L2(ω)≤C(|u|H1(0,T;L2(𝒪))+ε1/2|g|H1(ω)).
On the other hand, we prove that this estimate implies a regional controllability result for the same equation but with a control acting in 𝒪×(0,T) through the right-hand side: for any fixed f∈L2(ω) , the L2-norm of the control needed to have |u(T)|ω−f|H−1(ω)≤γ remains bounded with respect to γ if ε≤Cγ2.
