An interior discontinuity of a nonlinear elliptic-hyperbolic system

A nonlinear elliptic-hyperbolic system of partial differential equations which is a simplified form of the equations of viscous, compressible, barotropic flow at steady-state is studied. A boundary value problem for the system on a strip $D = (0,\alpha ) \times ( - \infty ,\infty )$ is considered. Zero boundary conditions for the velocities u and v on the sides of the strip D are imposed, and for pressure $p(0,y) = p_0 (y)$ is imposed, where $p_0 (y)$ has a jump at $y = 0$. Jump conditions for the system show that u and v are continuous. However, their derivatives and pressure have a curve of discontinuity. With sufficient small width of the strip D the Schauder fixed-point theorem gives a solution with a curve of discontinuity. The results suggest that there are two-dimensional, viscous, barotropic, steady-state compressible flows with discontinuity.