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Self-similar imploding solutions of the relativistic Euler equations

Feng Shao, Dongyi Wei, Zhifei Zhang

Abstract

Motivated by recent breakthrough on smooth imploding solutions of compressible Euler, we construct self-similar smooth imploding solutions of isentropic relativistic Euler equations with isothermal equation of state $p=\frac1\ell\varrho$ for \textit{all} $\ell>1$ in physical space dimension $d=2,3$ and for $\ell>1$ close to 1 in higher dimensions. This work is a crucial step toward solving the long-standing problem: finite time blow-up of the supercritical defocusing nonlinear wave equation.

Self-similar imploding solutions of the relativistic Euler equations

Abstract

Motivated by recent breakthrough on smooth imploding solutions of compressible Euler, we construct self-similar smooth imploding solutions of isentropic relativistic Euler equations with isothermal equation of state for \textit{all} in physical space dimension and for close to 1 in higher dimensions. This work is a crucial step toward solving the long-standing problem: finite time blow-up of the supercritical defocusing nonlinear wave equation.
Paper Structure (19 sections, 52 theorems, 471 equations, 2 figures, 1 table)

This paper contains 19 sections, 52 theorems, 471 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The self-similar equation and phase portrait
  3. Construction of $P_0-P_1$ curve and renormalization
  4. The $P_0-P_1$ curve
  5. Renormalization
  6. The $Q_0-Q_1$ curve
  7. Local smooth solution around the sonic point
  8. Slope at $Q_1$
  9. Eigenvalues
  10. Analytic solutions near $Q_1$
  11. Qualitative properties of local analytic solution
  12. Solution curve passing through the sonic point
  13. Global extension of the solution curve
  14. Barrier functions
  15. Global extension of the $Z-v$ solution curve
  16. ...and 4 more sections

Key Result

Theorem 1.1

For $k=d-1\geq 1$, there exists a critical $\ell_*(k)\in(1, +\infty]$ (see Eq.ell_*) such that for all $1<\ell<\ell_*(k)$, we can find a $\beta\in(0, k)$ such that the relativistic Euler equations Eq.relativistic_Euler with Eq.state admit a spherically symmetric self-similar solution defined on $[0, with the asymptotics for some real constants $\varrho_*>0, u^0_*\in\mathbb R\setminus\{0\}, u_*\in

Figures (2)

  • Figure 1: Phase portrait for the $Z-v$ system, with $m=1, \ell=2$ and $k=3$.
  • Figure 2: Phase portrait for the $z-u$ system, with $m=1.5, \ell=1.21$ and $k=3$.

Theorems & Definitions (114)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • Remark 3.5
  • ...and 104 more