Determination of convection terms and quasi-linearities appearing in diffusion equations

Abstract : We consider the highly nonlinear and ill-posed inverse problem of determining some general expression F (x, t, u, xu) appearing in the diffusion equation ∂tu − ∆xu + F (x, t, u, xu) = 0 on Ω × (0, T), with T > 0 and Ω a bounded open subset of R n , n 2, from measurements of solutions on the lateral boundary ∂Ω × (0, T). We consider both linear and nonlinear expression of F (x, t, u, xu). In the linear case, the equation is a convection-diffusion equation and our inverse problem corresponds to the unique recovery, in some suitable sense, of a time evolving velocity field associated with the moving quantity as well as the density of the medium in some rough setting described by non-smooth coefficients on a Lipschitz domain. In the nonlinear case, we prove the recovery of more general quasilinear expression appearing in a non-linear parabolic equation. Our result give a positive answer to the unique recovery of a general vector valued first order coefficient, depending on both time and space variable, and to the unique recovery inside the domain of quasilinear terms, from measurements restricted to the lateral boundary, for diffusion equations.
Type de document :
Pré-publication, Document de travail
2019
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https://hal.archives-ouvertes.fr/hal-02002056
Contributeur : Yavar Kian <>
Soumis le : jeudi 31 janvier 2019 - 14:35:31
Dernière modification le : lundi 4 février 2019 - 11:29:08

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  • HAL Id : hal-02002056, version 1

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Pedro Caro, Yavar Kian. Determination of convection terms and quasi-linearities appearing in diffusion equations. 2019. 〈hal-02002056〉

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