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

By Sergio Guerrero, A. Mercado and A. Osses

Abstract

In this paper we prove an inverse inequality for the parabolic equation υt‒ ϵΔυ+M.∇υ=ƒ1ω with Dirichlet boundary conditions. With the motivation of finding an estimate of ƒ 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 and O, and the time T , we are able to prove that, in the particular case when ƒ ∈ H 10(ω) and it does not depend on time, we have: | ƒ |L2(ω) ≤ C ( |u|H1(0,T ;L2O)) + ϵ1/2 | ƒ | 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 g ∈ L 2(ω),the L 2- norm of the control needed to have | u (T)|ω ‒ g | H -1 (ω) ≤ γ remains, bounded with respect to γ si ϵ ≤ C γ2

Topics: [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA]
Publisher: HAL CCSD
Year: 2006
OAI identifier: oai:HAL:hal-00113463v1
Provided by: Hal-Diderot
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