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Complex Dynamics of Wave-Character Transitions in Radially Symmetric Isentropic Euler Flows: Theory and Numerics

Eduardo Abreu, Geng Chen, Faris El-Katri, Erivaldo Lima

Abstract

We investigate the qualitative dynamics of smooth solutions to the radially symmetric isentropic compressible Euler equations, focusing specifically on the evolution of rarefactive and compressive wave characters across three distinct configurations: the outward supersonic, subsonic, and inward supersonic regimes. For each case, we establish structural restrictions on wave-character transitions and identify invariant sign domains for gradient variables under specific initial data conditions. While our findings refine existing invariance properties in the outward supersonic regime, they reveal novel asymmetric transition mechanisms in the subsonic and inward regimes that are absent in purely supersonic expanding cases. Consequently, we derive sufficient conditions for finite-time singularity formation. To complement the analytical results where closed-form solutions are unavailable, we provide numerical experiments using a Semi-Discrete Lagrangian-Eulerian (SDLE) formulation. These simulations reproduce the predicted wave-character dynamics and offer qualitative evidence that supports our theoretical findings, providing a unified description of wave transitions in radially symmetric isentropic gas dynamics.

Complex Dynamics of Wave-Character Transitions in Radially Symmetric Isentropic Euler Flows: Theory and Numerics

Abstract

We investigate the qualitative dynamics of smooth solutions to the radially symmetric isentropic compressible Euler equations, focusing specifically on the evolution of rarefactive and compressive wave characters across three distinct configurations: the outward supersonic, subsonic, and inward supersonic regimes. For each case, we establish structural restrictions on wave-character transitions and identify invariant sign domains for gradient variables under specific initial data conditions. While our findings refine existing invariance properties in the outward supersonic regime, they reveal novel asymmetric transition mechanisms in the subsonic and inward regimes that are absent in purely supersonic expanding cases. Consequently, we derive sufficient conditions for finite-time singularity formation. To complement the analytical results where closed-form solutions are unavailable, we provide numerical experiments using a Semi-Discrete Lagrangian-Eulerian (SDLE) formulation. These simulations reproduce the predicted wave-character dynamics and offer qualitative evidence that supports our theoretical findings, providing a unified description of wave transitions in radially symmetric isentropic gas dynamics.
Paper Structure (13 sections, 7 theorems, 74 equations, 54 figures)

This paper contains 13 sections, 7 theorems, 74 equations, 54 figures.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Supersonic Regime and Characteristic Variables
  4. Initial Conditions
  5. Boundary Conditions
  6. Subsonic Regime
  7. Mathematical Analysis
  8. Case 1: supersonic expanding wave, when $0<c_1<c_2$
  9. Case 2: subsonic expanding wave, when $c_1<0<c_2$
  10. Case 3: supersonic inward wave, when $c_1<c_2<0$
  11. Numerical Scheme: Semi-Discrete Lagrangian-Eulerian Formulation
  12. Numerical Experiments
  13. Conclusions

Key Result

Theorem 3.1

Let $c_1(r,0), c_2(r,0) \in C^1(\Omega)$. Further suppose that $c_1(r,0) > 0$ for all $r\in\Omega$, and that $c_1(r,t), c_2(r,t) \in C^1(D(\Omega,T))$ for some $T>0$. Then, for all $(r,t) \in D(\Omega,T)$, we have that $c_1(r,t)\geq \min(c_1(r,0))$, and $c_2(r,t) \leq ||c_2(r,0)||_{L^\infty}$.

Figures (54)

  • Figure 1: The initial density is shown on the left and its time-evolved state on the center and on the right.
  • Figure 2: The initial velocity is shown on the left and its time-evolved state on the center and on the right.
  • Figure 3: The initial pressure is shown on the left and its time-evolved state on the center and on the right.
  • Figure 4: The initial sound speed is shown on the left and its time-evolved state on the center and on the right.
  • Figure 5: The initial invariant curve in $(u,h)-$plane is shown on the left and its time-evolved state on the center and on the right.
  • ...and 49 more figures

Theorems & Definitions (17)

  • Definition 3.1
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Definition 3.2
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • ...and 7 more