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Amplitude blowup in compressible Euler flows without shock formation

Helge Kristian Jenssen

TL;DR

This work proves the existence of radially symmetric self-similar solutions to the 3D compressible Euler equations for an ideal gas (γ>1) that blow up at the center while remaining continuous elsewhere, and intriguingly can do so without forming shocks. By employing a self-similar ansatz with x = t/r^λ and reducing to a planar ODE system in the similarity variables (V,C), the authors analyze a rich array of critical points (P1–P9 and infinity) and construct a global, shock-free trajectory that passes through the origin and connects high- and low-state nodes, yielding a globally defined, continuous flow with amplitude blowup at t=0. A key outcome is that the isentropic constraint κ = ar{ ext{kappa}}(γ,λ) emerges from physicality constraints, and blowup can occur even when all fluid particles move toward the center at collapse, challenging conventional expectations about shock formation. The paper also discusses the potential for non-uniqueness via Hugoniot considerations, though in the isentropic setting such scenarios remain inconclusive, underscoring subtlety in multidimensional shock formation and continuation.

Abstract

Recent works have demonstrated that continuous self-similar radial Euler flows can drive primary (non-differentiated) flow variables to infinity at the center of motion. Among the variables that blow up at collapse is the pressure, and it is unsurprising that this type of behavior can generate an outgoing shock wave. In this work we prove that there is an alternative scenario in which an incoming, continuous 3-d flow suffers blowup, including in pressure, and yet remains continuous beyond collapse. We verify that this behavior is possible even in cases where the fluid is everywhere moving toward the center of motion at time of collapse. The results underscore the subtlety of shock formation in multi-dimensional flow.

Amplitude blowup in compressible Euler flows without shock formation

TL;DR

This work proves the existence of radially symmetric self-similar solutions to the 3D compressible Euler equations for an ideal gas (γ>1) that blow up at the center while remaining continuous elsewhere, and intriguingly can do so without forming shocks. By employing a self-similar ansatz with x = t/r^λ and reducing to a planar ODE system in the similarity variables (V,C), the authors analyze a rich array of critical points (P1–P9 and infinity) and construct a global, shock-free trajectory that passes through the origin and connects high- and low-state nodes, yielding a globally defined, continuous flow with amplitude blowup at t=0. A key outcome is that the isentropic constraint κ = ar{ ext{kappa}}(γ,λ) emerges from physicality constraints, and blowup can occur even when all fluid particles move toward the center at collapse, challenging conventional expectations about shock formation. The paper also discusses the potential for non-uniqueness via Hugoniot considerations, though in the isentropic setting such scenarios remain inconclusive, underscoring subtlety in multidimensional shock formation and continuation.

Abstract

Recent works have demonstrated that continuous self-similar radial Euler flows can drive primary (non-differentiated) flow variables to infinity at the center of motion. Among the variables that blow up at collapse is the pressure, and it is unsurprising that this type of behavior can generate an outgoing shock wave. In this work we prove that there is an alternative scenario in which an incoming, continuous 3-d flow suffers blowup, including in pressure, and yet remains continuous beyond collapse. We verify that this behavior is possible even in cases where the fluid is everywhere moving toward the center of motion at time of collapse. The results underscore the subtlety of shock formation in multi-dimensional flow.
Paper Structure (36 sections, 12 theorems, 141 equations, 2 figures)

This paper contains 36 sections, 12 theorems, 141 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result and outline
  3. Similarity variables and similarity ODEs
  4. Similarity ODEs and reduced similarity ODE
  5. Strategy and preliminary observations
  6. Strategy and an exact integral
  7. Preliminaries about critical points
  8. Critical points of the reduced similarity ODE
  9. Critical points on the $V$-axis
  10. Critical points off the $V$-axis
  11. $P_6$-$P_9$ in isentropic case
  12. Critical points at infinity
  13. Restricting $\lambda$ and $\kappa$ in terms of $n$ and $\gamma$
  14. Presence, location, and types of $P_4$-$P_9$
  15. Preliminaries
  16. ...and 21 more sections

Key Result

Theorem 1.1

Consider the 3-dimensional compressible Euler system (mass_m_d_full_eul)-(energy_m_d_full_eul) for an ideal gas (pressure1) with adiabatic index $\gamma>1$, and let the similarity parameter $\kappa$ have the "isentropic" value $\bar{\kappa}=-\frac{2(\lambda-1)}{\gamma-1}$. Then, for any $\lambda>1$

Figures (2)

  • Figure 1: Schematic picture of the relative locations of the zero-level sets of $F$ and $G$ (solid thick curves), the primary direction (solid thin line), and the critical line $L_-=\{C=-1-V\}$ (dash-dot) near $P_9$. Also shown is the vertical asymptote $V=V_*$ of the zero-level of $G$ (thin dotted), as well as the unique $P_9P_{-\infty}$-trajectory $\Sigma'$ (thick dotted). The three thick arrows indicate the direction of flow for solutions to the original ODE system (\ref{['V_sim2']})-(\ref{['C_sim2']}) as $x>0$ increases.
  • Figure 2: Schematic picture showing the $\Sigma$ trajectory (thick long dashes) which leaves the origin $P_1$ vertically and connects $P_1$ to $P_9$. $\Sigma$ is bounded above and below by the parabolic barriers $\Pi_1$ and $\Pi_2$ (thin curves), respectively, before entering the eye-shaped region between $P_5$ and $P_9$. The two thick arrows indicate how solutions to the original ODE system (\ref{['V_sim2']})-(\ref{['C_sim2']}) cross the barriers as $x>0$ increases.

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.1
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • Remark 4.1
  • Proposition 4.2
  • Lemma 5.1
  • ...and 16 more