On stationary Navier-Stokes equations in the upper-half plane

We study the incompressible stationary Navier-Stokes equations in the upper-half plane with homogeneous Dirichlet boundary condition and non-zero external forcing terms. Existence of weak solutions is proved under a suitable condition on the external forces. Weak-strong uniqueness criteria based on various growth conditions at the infinity of weak solutions are also given. This is done by employing an energy estimate and a Hardy's inequality. Several estimates of stream functions are carried out and two density lemmas with suitable weights for the homogeneous Sobolev space on 2 dimensional space are proved.