Navier–Stokes Equations with Regularized Directional Boundary Condition

We consider the steady Navier–Stokes equations in a bounded domain \(\varOmega \subset {\mathbb R}^n\) , \(n=2,3\) , with mixed boundary conditions and a nonhomogeneous forcing term. Instead of the classical do-nothing (CDN) boundary condition, a regularized directional do-nothing (RDDN) condition depending on a parameter \(0 < \delta \ll 1\) is imposed on a portion of the boundary. In the remaining part of the boundary, we consider a nonhomogeneous Dirichlet condition. An auxiliary reference flow, which also works as a lifting of the Dirichlet boundary values, is used to define the RDDN condition. We prove existence and uniqueness of weak solutions to the Navier–Stokes equations with RDDN condition, under appropriate assumptions on the size of the data, which, however, are less restrictive when compared with the case of a CDN outflow condition. We present numerical experiments that illustrate the advantages of using the DDN and RDDN boundary conditions.