On the persistence of Hölder regular patches of density for the inhomogeneous Navier-Stokes equations
Raphaël Danchin; Xin Zhang
Journal de l'École polytechnique — Mathématiques, Volume 4  (2017), p. 781-811

In our recent work dedicated to the Boussinesq equations [15], we established the persistence of solutions with piecewise constant temperature along interfaces with Hölder 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,ε regularity propagate for all time. Our result follows from the conservation of Hölder regularity along vector fields moving with the flow. The proof of that latter result is based on commutator estimates involving para-vector 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.

Dans notre travail récent consacré aux équations de Boussinesq [15], on a établi la persistance de solutions avec température constante par morceaux le long d’interfaces à régularité höldérienne. On aborde ici la même question pour les équations de Navier-Stokes inhomogène satisfaites par un liquide visqueux incompressible à densité variable. On démontre que, dans le cas légèrement non homogène, les poches de densité avec régularité 𝒞 1,ε se propagent pour tout temps. Notre résultat est conséquence de la conservation de la régularité höldérienne le long des champs de vecteurs transportés par le flot de la solution. La preuve de ce dernier résultat repose sur des estimations de commutateur mettant en jeu des para-champs et des espaces de multiplicateurs. L’analyse est plus compliquée que dans [15], dans la mesure où le couplage entre les équations de la masse et de la vitesse dans les équations de Navier-Stokes inhomogène est quasilinéaire alors qu’il est linéaire pour les équations de Boussinesq.

Received : 2016-12-01
Accepted : 2017-06-09
Published online : 2017-06-16
DOI : https://doi.org/10.5802/jep.56
Classification:  35K59,  76D05
Keywords: Inhomogeneous Navier-Stokes equations, 𝒞 1,ε density patch, striated regularity
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     author = {Rapha\"el Danchin and Xin Zhang},
     title = {On the persistence of H\"older regular patches of density for the inhomogeneous Navier-Stokes equations},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {4},
     year = {2017},
     pages = {781-811},
     doi = {10.5802/jep.56},
     zbl = {06754341},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2017__4__781_0}
}
Danchin, Raphaël; Zhang, Xin. On the persistence of Hölder regular patches of density for the inhomogeneous Navier-Stokes equations. Journal de l'École polytechnique — Mathématiques, Volume 4 (2017) , pp. 781-811. doi : 10.5802/jep.56. https://jep.centre-mersenne.org/item/JEP_2017__4__781_0/

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