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We consider the stationary navier stokes equation with mixed boundary conditions $$ \begin{align} -\nu\Delta u +(u\cdot\nabla) u + \nabla p &= 0\ \textrm{in}\ \Omega \\ div\ u&=0\ \textrm{in}\ \Omega\\ u &=g\ \textrm{on} \ \Gamma_d \\ pn-\nu\nabla u\cdot n&=0\ \textrm{on}\ \Gamma_n \\ \partial \Omega &= \Gamma_d \stackrel{\cdot}{\cup} \Gamma_n \end{align} $$ i.e. No external forces $f$. Inhomogenous Dirichlet boundary Data and a "do nothing" condition on another part of the boundary. Are for that kind of boundary conditions any existence results for solutions?

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  • $\begingroup$ First of all, the free boundary condition should be $pn-\nu(\nabla u+(\nabla u)^T)n=0$. Second, what kind of existence results are you looking for? For small data, standard methods work. For large data, there is not much hope if there are free surfaces, as demonstrated by phenomena such as wave breaking, jet breakup etc. $\endgroup$ May 26, 2015 at 12:23
  • $\begingroup$ By looking in the literature i found the following paper global-sci.org/jcm/galley/JCM4347.pdf. They consider $f\neq 0$ and $g=0$ on $\Gamma_d$. Look at formula (2.7). They argue that they have no control over the inflow at the "do-nothing" boundary part, so that they run in trouble proofing existence. $\endgroup$
    – user74165
    May 27, 2015 at 12:06

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