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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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