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Influence des perturbations géométriques de domaines sur les solutions d'équations aux dérivées partielles

Abstract : In this thesis, we are interested in the influence of geometric perturbations of the boundaries of domains on the solutions to partial differential equations with vector values, by a geometric effect of the perturbations that is called the rugosity effect. This effect consists in transforming non penetration conditions imposed on a sequence of oscillating boundaries, that converges in a certain geometric sense to a smooth boundary, into a so-called friction-driven boundary condition, which was introduced by Bucur, Feireisl and Necasova in 2009. We characterize the rugosity effect produced by different types of boundaries, periodic or not, relying on the use of Young measures and capacitary measures, that allow us to understand the oscillations of the normal vectors by Gamma-convergence tools. We prove the stability of the trajectory of a deformable self-propelled solid, at low Reynolds number, with respect to the deformation vector field that is imposed on the solid, and propose a numerical method to solve the model in dimension 2. We study the well-posedness of the problem of the drag associated to a solid with friction-driven boundary conditions. Using the rugosity effect, we set the problem of drag minimization in terms of the micro-structure of the boundary, and prove that this problem admits a solution. We illustrate the theoretical results by numerical examples and carry out a numerical optimization algorithm.
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Contributor : Matthieu Bonnivard <>
Submitted on : Wednesday, June 8, 2011 - 10:31:00 AM
Last modification on : Friday, November 6, 2020 - 3:26:30 AM
Long-term archiving on: : Friday, September 9, 2011 - 12:03:42 PM


  • HAL Id : tel-00555121, version 2



Matthieu Bonnivard. Influence des perturbations géométriques de domaines sur les solutions d'équations aux dérivées partielles. Mathématiques [math]. Université de Grenoble, 2010. Français. ⟨tel-00555121v2⟩



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