Loss of Regularity of Solutions of the Lighthill Problem for Shock Diffraction for Potential Flow

TL;DR

This paper demonstrates that potential flow models cannot produce regular solutions for shock diffraction at a wedge corner, implying the need to use the full Euler system for accurate modeling.

Contribution

The paper proves the non-existence of subsonic regular solutions for the Lighthill problem in potential flow, highlighting the limitations of simplified models in shock diffraction scenarios.

Findings

No regular subsonic solutions exist at the wedge corner in potential flow.

Potential flow models are inadequate for the Lighthill shock diffraction problem.

Mathematical techniques developed can be applied to similar free boundary problems.

Abstract

We are concerned with the suitability of the main models of compressible fluid dynamics for the Lighthill problem for shock diffraction by a convex corned wedge, by studying the regularity of solutions of the problem, which can be formulated as a free boundary problem. In this paper, we prove that there is no regular solution that is subsonic up to the wedge corner for potential flow. This indicates that, if the solution is subsonic at the wedge corner, at least a characteristic discontinuity (vortex sheet or entropy wave) is expected to be generated, which is consistent with the experimental and computational results. Therefore, the potential flow equation is not suitable for the Lighthill problem so that the compressible Euler system must be considered. In order to achieve the non-existence result, a weak maximum principle for the solution is established, and several other mathematical techniques are developed. The methods and techniques developed here are also useful to the other problems with similar difficulties.