Abstract
We consider the stationary incompressible Navier–Stokes equation in the half-plane with inhomogeneous boundary condition. We prove the existence of strong solutions for boundary data close to any Jeffery–Hamel solution with small flux evaluated on the boundary. The perturbation of the Jeffery–Hamel solution on the boundary has to satisfy a nonlinear compatibility condition which corresponds to the integral of the velocity field on the boundary. The first component of this integral is the flux which is an invariant quantity, but the second, called the asymmetry, is not invariant, which leads to one compatibility condition. Finally, we prove the existence of weak solutions, as well as weak–strong uniqueness for small data and provide numerical simulations.
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Communicated by Nader Masmoudi.
This work was partially supported by the Swiss National Science Foundation Grants 124403, 140305, and 161996.
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Guillod, J., Wittwer, P. On the Stationary Navier–Stokes Equations in the Half-Plane. Ann. Henri Poincaré 17, 3287–3319 (2016). https://doi.org/10.1007/s00023-016-0470-0
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DOI: https://doi.org/10.1007/s00023-016-0470-0