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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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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DOI: https://doi.org/10.1007/s00205-012-0563-y