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

TL;DR

The paper addresses the regularity of self-similar, shock-bearing solutions to two-dimensional Riemann problems for the isentropic Euler system, introducing a general framework to analyze local regularity across canonical shock configurations. By regularizing the system and studying the transport of vorticity $\omega=\partial_1 v_2-\partial_2 v_1$, together with a renormalization argument on domains with boundary and DiPerna–Lions commutator estimates, it proves that, in the subsonic domain $\Omega$, it is generally impossible for both density $\rho$ and velocity $\mathbf{v}$ to belong to $H^1(\Omega)$. This low-regularity result implies that self-similar Riemann solutions with shocks can exhibit discontinuous velocity and more intricate structure than potential-flow analogues. The framework is then applied to regular shock reflection, Prandtl reflection, Lighthill diffraction, and four-shock interactions, showing the same $H^1$-incompatibility in each case and highlighting the pivotal role of vorticity in shaping multi-dimensional shock dynamics.

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 it is not possible that both the density and the velocity are 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 density and the velocity are 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.

Figures (9)

• Figure 3.1: The Riemann Problem in a General Setting
• Figure 4.1: Supersonic regular reflection
• Figure 4.2: Subsonic regular reflection
• Figure 4.3: Non-symmetric supersonic regular reflection
• Figure 4.4: Non-symmetric subsonic regular reflection

Theorems & Definitions (26)

• Definition 2.1
• Definition 2.2
• Definition 3.1
• Theorem 3.2: Main Theorem
• proof
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 4.1
• proof