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Converging/diverging self-similar shock waves: from collapse to reflection

Juhi Jang, Jiaqi Liu, Matthew Schrecker

Abstract

We solve the continuation problem for the non-isentropic Euler equations following the collapse of an imploding shock wave. More precisely, we prove that the self-similar Güderley imploding shock solutions for a perfect gas with adiabatic exponent $γ\in(1,3]$ admit a self-similar extension consisting of two regions of smooth flow separated by an outgoing spherically symmetric shock wave of finite strength. In addition, for $γ\in(1,\frac53]$, we show that there is a unique choice of shock wave that gives rise to a globally defined self-similar flow with physical state at the spatial origin.

Converging/diverging self-similar shock waves: from collapse to reflection

Abstract

We solve the continuation problem for the non-isentropic Euler equations following the collapse of an imploding shock wave. More precisely, we prove that the self-similar Güderley imploding shock solutions for a perfect gas with adiabatic exponent admit a self-similar extension consisting of two regions of smooth flow separated by an outgoing spherically symmetric shock wave of finite strength. In addition, for , we show that there is a unique choice of shock wave that gives rise to a globally defined self-similar flow with physical state at the spatial origin.
Paper Structure (14 sections, 42 theorems, 202 equations, 3 figures)

This paper contains 14 sections, 42 theorems, 202 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Collapsing shock solution and continuation problem
  3. Continuation through the origin
  4. Maximal extension
  5. Continuation of trajectories from $P_8$
  6. Continuation of trajectories from $P_6$
  7. Continuation of trajectories from $P_6$ case I: $z\in[z_s,z_M]$
  8. Continuation of trajectories from $P_6$ case II: $z\in(z_g,z_s)$
  9. Jump Locus and existence of $P_H$
  10. Proof of the Main Theorem \ref{['main thm']}
  11. Proof of Lemma \ref{['L:FG=0']}
  12. Quantitative Properties
  13. Applications of Fourier-Budan Theorem
  14. Polynomials

Key Result

Theorem 1.1

Let $\gamma\in(1,3]$, $m=1,2$, and take $\lambda=\lambda_{std}(\gamma,m)$. Let $(\rho_0,u_0,c_0)(r)$ given by eq:collapsingterminal be the terminal data of the collapsing Güderley shock solution. Then, for $t>0$, there exists a piecewise smooth entropy-admissible weak solution $(\rho,u,c)$ of symmet

Figures (3)

  • Figure 1: the solution trajectory when $m=1$, $\gamma = 1.5$ and $z = 0.14$.
  • Figure 2: $\gamma=2$, $z=0.119$, $m=2$
  • Figure 3: $\gamma=3$, $z=z_M(3)$, $m=2$

Theorems & Definitions (80)

  • Theorem 1.1
  • Theorem 1.2: JLS23
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 1.6
  • proof
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • ...and 70 more