Constraints on possible singularities for the unsteady transonic small disturbance (UTSD) equations

We discuss the singular behavior of solutions to two-dimensional, general second-order, uniformly elliptic equations in divergence form, with bounded measurable coefficients and discontinuous Dirichlet data along a portion of a Lipschitz boundary. We show that the conjugate to the solution develops a singularity that is at least logarithmic along the boundary at the points of discontinuity in the boundary data. A problem like this arises in the study of self-similar solutions to the hyperbolic conservation laws in two space dimensions given by the unsteady transonic small disturbance (UTSD) flow equations. These solutions model the reflection of a weak shock wave upon a thin wedge in the regime where the von Neumann paradox applies. The present result provides a step in the direction of understanding the nature of the solutions to the UTSD equations near a triple point. It shows that the flow behind the triple point cannot be strictly subsonic under some mild assumptions on the solutions.