Discontinuous Solutions of Steady State, Viscous Compressible Navier-Stokes Equations

We study a steady-state, viscous, compressible Navier-Stokes flow in a rectangle Ω ≡ (0, 1) × (−1, 1) with the boundary condition (u, v) = (1, 0) for the velocity field (u, v) and the condition p(0, y) = p0(y) for the pressure p on {0} × (−1, 1), which is the part of the boundary where the stream lines emanate. Under the condition that p0(y) has a jump at y = 0, we establish the existence and uniqueness of the solution having discontinuity along the stream line starting from the origin.