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# Étude asymptotique de deux problèmes de la mécanique de fluides : formation de couches limites près des côtes rugueuses non périodiques et comportement d’ondes progressives dans le problème de Hele-Shaw

Abstract : This manuscript deals with the asymptotic analysis of two problems arising in fluid mechanics: the effect of roughness on oceanic motion taking as a starting point the single-layered quasigeostrophic equation and the mathematical description of congestion phenomena in tumor growth. First, we are interested in the impact of the irregularities of the coastline on wind-driven oceanic motion when the geometry of the coasts does not follow a specific spatial pattern. The assumption on the roughness has two main consequences in the asymptotic analysis of the quasigeostrophic model: the governing boundary layer equations are defined in infinite domains with not-decaying boundary data, and the eastern boundary layer exhibits convergence issues far from the boundary. We establish the well-posedness of the solution of the boundary layer profiles in nonlocalized Sobolev space by adding ergodicity properties and using pseudo-differential analysis. We construct an approximate solution to the original problem and analyze its convergence. In the second part of this work, we study a one-dimensional porous medium equation (PME) modeling the mechanical properties of tumor growth. We are interested in the singular stiff pressure law'' limit when the PME degenerates towards a free boundary problem of Hele-Shaw type. We provide a refined description of the traveling waves in the vicinity of the transition between the free domain with zero pressure and the congested domain with positive pressure and then perform a stability analysis of the traveling waves.
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Submitted on : Wednesday, May 25, 2022 - 4:47:26 PM
Last modification on : Thursday, May 26, 2022 - 3:48:27 AM

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• HAL Id : tel-03678970, version 1

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Gabriela López Ruiz. Étude asymptotique de deux problèmes de la mécanique de fluides : formation de couches limites près des côtes rugueuses non périodiques et comportement d’ondes progressives dans le problème de Hele-Shaw. Equations aux dérivées partielles [math.AP]. Sorbonne Université, 2021. Français. ⟨NNT : 2021SORUS347⟩. ⟨tel-03678970⟩

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