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On the persistence of Hölder regular patches of density for the inhomogeneous Navier-Stokes equations

Xin Zhang 1 Raphaël Danchin 1 
1 UMR8050
LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
Abstract : In our recent work dedicated to the Boussinesq equations [15], we established the persistence of solutions with piecewise constant temperature along interfaces with Holder regularity. We here address the same question for the inhomogeneous Navier-Stokes equations satisfied by a viscous incompressible and inhomogeneous fluid. We prove that, indeed, in the slightly inhomogeneous case, patches of densities with C-1,C-epsilon regularity propagate for all time. Our result follows from the conservation of Holder regularity along vector fields moving with the flow. The proof of that latter result is based on commutator estimates involving paravector fields, and multiplier spaces. The overall analysis is more complicated than in [15], since the coupling between the mass and velocity equations in the inhomogeneous Navier-Stokes equations is quasilinear while it is linear for the Boussinesq equations.
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Submitted on : Sunday, October 8, 2017 - 1:10:43 PM
Last modification on : Saturday, January 15, 2022 - 4:13:14 AM

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Xin Zhang, Raphaël Danchin. On the persistence of Hölder regular patches of density for the inhomogeneous Navier-Stokes equations. Journal de l'École polytechnique — Mathématiques, École polytechnique, 2017, 4, pp.781-811. ⟨10.5802/jep.56⟩. ⟨hal-01612798⟩

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