An Inverse Problem for Steady Supersonic Potential Flow Past a Bending Wall

We study an inverse problem of determining the shape of a bending wall with a given surface pressure distribution in the two-dimensional steady supersonic potential flow. The given pressure distribution on the wall surface is assumed to be a small perturbation of the pressure distribution corresponding to a bending convex wall and to have a bounded total variation. In this setting, we first give the background solution which only contains strong rarefaction waves generated by a bending convex wall. Then, we construct the approximate boundaries and corresponding approximate solutions of the inverse problem within a perturbation domain of this background solution. To achieve this, we employ a modified wave-front tracking algorithm. Finally, we show that the limit of approximate solutions provides a global entropy solution for the inverse problem, and the limit of approximate boundaries gives a boundary profile representing the shape of a bending wall that yields the given pressure distribution.