Affiliations: Université Pierre et Marie Curie-Paris 6, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005 France E-mail: [email protected] | Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170/3—Correo 3, Santiago, Chile Laboratoire de Mathematiques, Université de Versailles-Saint-Quentin, 45 avenue des Etats-Unis, 78000 Versailles, France E-mail: [email protected] | Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170/3—Correo 3, Santiago, Chile and Centro de Modelamiento Matemático, UMI 2807 CNRS-Uchile, Chile E-mail: [email protected]
Note: [] Supported by grant MECESUP 0009.
Note: [] Supported by grants ECOS C04E08 and FONDECYT 1061263.
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))+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.
