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On the Flux Problem in the Theory of Steady Navier–Stokes Equations with Nonhomogeneous Boundary Conditions

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Abstract

We study the nonhomogeneous boundary value problem for Navier–Stokes equations of steady motion of a viscous incompressible fluid in a two-dimensional, bounded, multiply connected domain \({\Omega = \Omega_1 \backslash \overline{\Omega}_2, \overline\Omega_2\subset \Omega_1}\) . We prove that this problem has a solution if the flux \({\mathcal{F}}\) of the boundary value through ∂Ω 2 is nonnegative (inflow condition). The proof of the main result uses the Bernoulli law for a weak solution to the Euler equations and the one-sided maximum principle for the total head pressure corresponding to this solution.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Acad. Koptyug pr. 4, Novosibirsk, Russia

    Mikhail V. Korobkov

  2. Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, 03225, Vilnius, Lithuania

    Konstantin Pileckas

  3. Dipartimento di Matematica, Seconda Università di Napoli, via Vivaldi 43, Caserta, 81100, Italy

    Remigio Russo

Authors
  1. Mikhail V. Korobkov
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  2. Konstantin Pileckas
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  3. Remigio Russo
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Correspondence to Mikhail V. Korobkov.

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Communicated by V. Šverák

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Korobkov, M.V., Pileckas, K. & Russo, R. On the Flux Problem in the Theory of Steady Navier–Stokes Equations with Nonhomogeneous Boundary Conditions. Arch Rational Mech Anal 207, 185–213 (2013). https://doi.org/10.1007/s00205-012-0563-y

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