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arXiv · 2504.07034

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

Gui-Qiang G. Chen·Mikhail Feldman·Wei Xiang
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Abstract

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 $H^1$ 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.

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Gui-Qiang G. Chen, Mikhail Feldman, Wei Xiang. 2025-04-09. Low Regularity of Self-Similar Solutions of Two-Dimensional Riemann problems with Shocks for the Isentropic Euler system. https://arxiv.org/abs/2504.07034

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