An inverse inequality for some transport–diffusion equation. Application to the regional approximate controllability

Abstract

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))}+ε^1/2|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.