Singularity formation for the supersonic inward wave of compressible Euler equations with radial symmetry

TL;DR

The paper addresses finite-time singularity formation for smooth solutions of the radially symmetric compressible Euler equations with a polytropic gas ($p=K\rho^\gamma$), focusing on inward supersonic waves. It advances the analysis by using the characteristic method and an invariant-domain framework, introducing weighted gradient variables $\tilde{\alpha}$ and $\tilde{\beta}$ to obtain a homogeneous, decoupled Riccati-type structure and to control an invariant region for the flow. The main result shows that for $\gamma \ge 3$, sufficiently large negative initial $\tilde{\beta}_0$ drives $\tilde{\beta}$ to blow up in finite time along a 1-characteristic, yielding a gradient singularity while the solution remains within the invariant region up to blow-up. This extends singularity formation results from expanding to inward radially symmetric waves and highlights the role of geometric effects and the gradient dynamics in focusing and shock formation.

Abstract

In this paper, we consider the singularity formation of smooth solutions for the compressible radially symmetric Euler equations. By applying the characteristic method and the invariant domain idea, we show that, for polytropic ideal gases with $γ\geq3$, the smooth solution develops a singularity in finite time for a class of initial supersonic inward waves.

Figures (2)

• Figure 1: The region $\Omega_T$.
• Figure 2: (a) The region of $(\tilde{\alpha},\tilde{\beta})$ for $(r,t)\in\Omega_T$; (b) The region of $(h,u)$ for $(r,t)\in\Omega_T$.

Theorems & Definitions (8)

• Lemma 2.1: Lemmas 3.1 CEHS
• Remark 1
• Remark 2
• Lemma 1
• Remark 3
• proof
• Theorem 1
• proof