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Existence and singularity formation for the supersonic expanding wave of radially symmetric non-isentropic compressible Euler equations

Geng Chen, Faris A. El-Katri, Yanbo Hu

Abstract

This paper studies the existence and singularity formation of supersonic expanding waves for the radially symmetric non-isentropic compressible Euler equations of polytropic gases. We introduce a suitable pair of gradient variables to characterize the rarefaction and compression properties of the solutions. Based on their Riccati equations, we construct several useful invariant domains to establish a series of priori estimates of solutions under some assumptions on the initial data. We show that the solution is smooth in the characteristic triangle or quadrangle domain if both of these two gradient variables are non-negative at the initial time. On the other hand, when one of these two variables is very negative at some initial point, the solution forms a singularity in finite time.

Existence and singularity formation for the supersonic expanding wave of radially symmetric non-isentropic compressible Euler equations

Abstract

This paper studies the existence and singularity formation of supersonic expanding waves for the radially symmetric non-isentropic compressible Euler equations of polytropic gases. We introduce a suitable pair of gradient variables to characterize the rarefaction and compression properties of the solutions. Based on their Riccati equations, we construct several useful invariant domains to establish a series of priori estimates of solutions under some assumptions on the initial data. We show that the solution is smooth in the characteristic triangle or quadrangle domain if both of these two gradient variables are non-negative at the initial time. On the other hand, when one of these two variables is very negative at some initial point, the solution forms a singularity in finite time.
Paper Structure (9 sections, 11 theorems, 158 equations, 1 figure)

This paper contains 9 sections, 11 theorems, 158 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The equations, assumptions and main results
  3. The characteristic equations
  4. The assumptions and main results
  5. The Riccati equations of $(\alpha,\beta)$
  6. $C^1$-estimates of the solution
  7. Proof of the theorems
  8. Proof of Theorem \ref{['thm1']}
  9. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1

Let Assumptions asu1-asu4 be satisfied. Then, for the initial data $(\rho_0(r), u_0(r), S_0(r))$, the radially symmetric non-isentropic Euler equations 1.1 with 1.2 admits a smooth solution $(\rho, u, S)(r,t)$ on the entire domain $\Omega_d$. Furthermore, the solution fulfils for some positive constant $\widetilde{\mathcal{C}}_1> \mathcal{C}_1$.

Figures (1)

  • Figure 1: The domain $\Omega_d$. When $T_0\geq T_m$ (Left), $\Omega_d$ is the characteristic triangle domain generated by interval $[b_1,b_2]$. When $T_0< T_m$ (Right), $\Omega_d$ is the characteristic quadrangle domain generated by interval $[b_1,b_2]$. Here $T_m$ is the intersection time of the 3-characteristic originating from point $(b_1,0)$ and the 1-characteristic originating from point $(b_2,0)$.

Theorems & Definitions (30)

  • Definition 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Theorem 2
  • Remark 5
  • Lemma 3.1
  • proof
  • ...and 20 more