Méthodes d'analyse de Fourier en hydrodynamique : des mascarets aux fluides avec capillarité

Cosmin Burtea 1
1 UMR8050
LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
Abstract : The first part of the present thesis deals with the so -called abcd systems which were derived by J.L. Bona, M. Chen and J.-C. Saut back in 2002. These systems are approximation models for the waterwaves problem in the Boussinesq regime, that is, waves of small amplitude and long wavelength. In the first two chapters we address the long time existence problem which consists in constructing solutions for the Cauchy problem associated to the abcd systems and prove that the maximal time of existence is bounded from below by some physically relevant quantity. First, we consider the case of initial data belonging to some Sobolev spaces imbedded in the space of continuous functions which vanish at infinity. Physically, this corresponds to spatially localized waves. The key ingredient is to construct a nonlinear energy functional which controls appropriate Sobolev norms on the desired time scales. This is accomplished by working with spectrally localized equations. The two important features of our method is that we require lower regularity levels in order to develop a long time existence theory and we may treat in an unified manner most of the cases corresponding to the different values of the parameters. In the second chapter, we prove the long time existence results for the case of data that does not necessarily vanish at infinity. This is especially useful if one has in mind bore propagation. One of the key ideas of the proof is to consider a well-adapted high-low frequency decomposition of the initial data. In the third chapter, we propose finite-volume schemes in order to construct numerical solutions. We use these schemes in order to study traveling waves interaction. The second part of this manuscript, is devoted to the study of optimal regularity issues for the incompressible inhomogeneous Navier-Stokes system and the Navier-Stokes-Korteweg system used in order to take in account capillarity effects. More precisely, we prove that these systems are well-posed in their truly critical spaces i.e. the spaces that have the same scale invariance as the system itself. In order to achieve this we derive new estimates for a Stoke-like problem with time independent variable coefficients.
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Cosmin Burtea. Méthodes d'analyse de Fourier en hydrodynamique : des mascarets aux fluides avec capillarité. Equations aux dérivées partielles [math.AP]. Université Paris-Est, 2017. Français. ⟨tel-01563716⟩

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