Variational resolution of outflow boundary conditions for incompressible Navier-Stokes

This paper focuses on the so-called weighted inertia-dissipation-energy variational approach for the approximation of unsteady Leray-Hopf solutions of the incompressible Navier-Stokes system. Initiated in (Ortiz et al 2018 Nonlinearity 31 5664-82), this variational method is here extended to the case of non-Newtonian fluids with power-law index r >= 11/5 in three space dimension and large nonhomogeneous data. Moreover, boundary conditions are not imposed on some parts of boundaries, representing, e.g., outflows. Correspondingly, natural boundary conditions arise from the minimisation. In particular, at walls we recover boundary conditions of Navier-slip type. At outflows and inflows, we obtain the condition -1/2 vertical bar v vertical bar(2) n + Tn = 0. This provides the first theoretical explanation for the onset of such boundary conditions.