Low Regularity of Self-Similar Solutions of Two-Dimensional Riemann Problems with Shocks for the Isentropic Euler System

Journal article

We are concerned with the low regularity of self-similar solutions of two-dimensional Riemann problems for the isentropic Euler system. We establish a general framework for the analysis of the local regularity of such solutions for a class of two-dimensional Riemann problems for the isentropic Euler system, which includes the regular shock reflection problem, the Prandtl reflection problem, the Lighthill diffraction problem, and the four-shock Riemann problem. We prove that the velocity is not in in the subsonic domain for the self-similar solutions of these problems in general. This indicates that the self-similar solutions of the Riemann problems with shocks for the isentropic Euler system are of much more complicated structure than those for the Euler system for potential flow; in particular, the velocity is not necessarily continuous in the subsonic domain. The proof is based on a regularization of the isentropic Euler system to derive the transport equation for the vorticity, a renormalization argument extended to the case of domains with boundary, and DiPerna-Lions-type commutator estimates.

Authors: Chen, GG; Feldman, M; Xiang, W

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Annals of PDE
• Volume: 12
• Issue: 1
• Article number: 8
• Publication date: 2026-03-02
• Acceptance date: 2025-10-28
• DOI: 10.1007/s40818-025-00225-z
• EISSN: 2199-2576
• ISSN: 2524-5317
• Language: English
• Keywords: 76N10; Transonic shock; Prandtl reflection; Nonlinear; Compressible flow; Fine properties; Low regularity; 35L67; 76G25; 35B36; Self-similar; 76H05; Potential flow; 35B45; Shock reflection-diffraction; 35B40; 35L04; 35L65; 35L03; 76J20; Secondary: 35R35; 35J70; 35B65; Conservation laws; Lighthill diffraction; 76N20; Free boundary; Regular shock reflection; 35J67; 76L05; 35M12; Primary: 35M10