Méthodes de moyennisation stroboscopique appliquées aux équations aux dérivées partielles hautement oscillantes

Abstract : This thesis presents some original work in the field of high order averaging procedure. In particular, we are interested in stroboscopic and quasi-stroboscopic averaging procedure in abstract Banach or Hilbert spaces. This procedures is applied to concrete examples: some highly oscillatory evolution equations. More precisely, we first show a theorem of stroboscopic averaging in a Banach space where we obtain exponential error estimates. This theorem is then applied on two semi-linear and highly oscillatory wave equations. We also put in evidence that the {\it Stroboscopic Averaging Method} works fine with a semi-linear wave equation with Dirichlet conditions. Finally, the averaging procedure puts in evidence, numerically, an interesting dynamics regarding the semi-linear wave equation with Dirichlet conditions. In a second part, we present a quasi-stroboscopic averaging theorem in a Hilbert space with exponential error estimates. This theorem is applied on a semi-linear Schrödinger equation. This equation has first, to be project in a finite dimensional space in order to fit in the hypotheses of the theorem. We then write a quasi-stroboscopic averaging theorem for a semi-linear Schrödinger equation with polynomial error estimates.
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Submitted on : Friday, June 3, 2016 - 1:42:09 PM
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Guillaume Leboucher. Méthodes de moyennisation stroboscopique appliquées aux équations aux dérivées partielles hautement oscillantes. Equations aux dérivées partielles [math.AP]. Université Rennes 1, 2015. Français. ⟨NNT : 2015REN1S121⟩. ⟨tel-01326270⟩

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