Qualitative study of solutions of the system of Navier-Stokes equations with variable density

Abstract : This thesis is dedicated to two different problems in the mathematical study of the viscous incompressible fluids: the persistence of tangential regularity and the motion of a free surface.The first problem concerns the study of the qualitative properties of some thermodynamical quantities in incompressible fluid models, such as the temperature for Boussinesq system with no diffusion and the density for the non-homogeneous Navier-Stokes system. Typically, we assume those two quantities to be initially piecewise constant along an interface with H"older regularity.As a consequence of stability of certain directional smoothness of the velocity field, we establish that the regularity of the interfaces persist globally with respect to time both in the two dimensional and higher dimensional cases (under some smallness condition). Our strategy is borrowed from the pioneering works by J.-Y.Chemin in 1990s on the vortex patch problem for ideal fluids.Let us emphasize that, apart from the directional regularity, we only impose rough (critical) regularity on the velocity field. The proof requires tools from para-differential calculus and multiplier space theory.In the last part of this thesis, we are concerned with the free boundary value problem for two-phase density-dependent Navier-Stokes system.This model is used to describe the motion of two immiscible liquids, like the oil and the water. Such mixture may occur in different situations, such as in a fixed bounded container, in a moving bounded droplet or in a river with finite depth. We establish the short time well-posedness for this problem. Our result strongly relies on the $L_p$-$L_q$ maximal regularity theoryfor parabolic equations
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Xin Zhang. Qualitative study of solutions of the system of Navier-Stokes equations with variable density. Numerical Analysis [math.NA]. Université Paris-Est, 2017. English. ⟨NNT : 2017PESC1215⟩. ⟨tel-01762929⟩

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