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Incompressible Flows with Piecewise Constant Density

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Abstract

We investigate the incompressible Navier–Stokes equations with variable density. The aim is to prove existence and uniqueness results in the case of discontinuous initial density. In dimension n = 2,3, assuming only that the initial density is bounded and bounded away from zero, and that the initial velocity is smooth enough, we get the local-in-time existence of unique solutions. Uniqueness holds in any dimension and for a wider class of velocity fields. In particular, all those results are true for piecewise constant densities with arbitrarily large jumps. Global results are established in dimension two if the density is close enough to a positive constant, and in n dimensions if, in addition, the initial velocity is small. The Lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper.

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Authors and Affiliations

  1. Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, IUF, 61 Avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Raphaël Danchin

  2. 2 Instytut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski, ul. Banacha, 202-097, Warszawa, Poland

    Piotr BogusŁaw Mucha

Authors
  1. Raphaël Danchin
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  2. Piotr BogusŁaw Mucha
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Correspondence to Raphaël Danchin.

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Communicated by L. Saint-Raymond

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Danchin, R., Mucha, P.B. Incompressible Flows with Piecewise Constant Density. Arch Rational Mech Anal 207, 991–1023 (2013). https://doi.org/10.1007/s00205-012-0586-4

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