Loss of regularity of solutions of the lighthill problem for shock diffraction for potential flow

Journal article

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 nonexistence 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.

Authors: Chen, G-Q; Feldman, M; Hu, J; Xiang, W

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial & Applied Mathematics
• Journal: SIAM Journal on Mathematical Analysis
• Volume: 52
• Issue: 2
• Pages: 1096-1114
• Publication date: 2020-03-12
• Acceptance date: 2020-01-02
• DOI: 10.1137/19m1284531
• EISSN: 1095-7154
• ISSN: 0036-1410
• Language: English
• Keywords: degenerate eliptic equations; mixed eliptic-hyperbolic type; compressible flow; loss of regularity; lighthill problem; potential flow equation; free boundary problems; FFR; conservation laws; shock diffraction; nonlinear equations of second order