The unsteady transonic small disturbance equation: Data on oblique curves

Abstract

We propose and solve a new problem for the unsteady transonic small disturbance equation. Data are given for the self-similar equation in a fixed, bounded region of similarity space, where on a part of the boundary the equation has degenerate type (a `sonic line') and on the remainder it is elliptic. Previous results on this problem have chosen data so that the solution is constant on the sonic line, but we set up a situation where the solution is not constant on the sonic part of the boundary. The solution we find is Lipschitz up to the boundary. Our solution sets the stage for resolution of some interesting Riemann problems for this equation and for other multidimensional conservation laws.