Variational resolution of outflow boundary conditions for incompressible Navier-Stokes

This paper focuses on the so-called Weighted Inertia-Dissipation-Energy (WIDE) variational approach for the approximation of unsteady Leray-Hopf solutions of the incompressible Navier-Stokes system. Initiated in [56], this variational method is here extended to the case of non-Newtonian fluids with power-law index $r\geq11/5$ in three space dimensions 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 minimization. In particular, at walls we recover boundary conditions of Navier-slip type. At outflows and inflows, we obtain the condition $-\frac12|\boldsymbol{v}|^2\boldsymbol{n}+\mathbb{T}\boldsymbol{n}=0$. This provides the first theoretical explanation for the onset of such boundary conditions.