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Local and Global Results on Three Dimensional Rarefaction Waves in Spherical Symmetry

Ruotong Zhang

TL;DR

The paper constructs and analyzes centered rarefaction waves for three-dimensional compressible Euler flow with spherical symmetry, addressing both local existence from general exterior backgrounds and global existence for backgrounds close to constant states with decay. The authors deploy an acoustical-geometry framework, using Riemann invariants and a Cauchy–characteristic approach to propagate data along outgoing null cones, and derive robust energy–flux estimates to control solutions. They establish precise boundary-data propagation, construct local approximate solutions, and prove convergence to genuine rarefaction waves, followed by a global extension under small perturbations of a constant state with decay, including detailed regularity propagation. The results advance understanding of multidimensional rarefaction waves in curved geometries and provide techniques that may inform the full non-symmetric 3D problem and related exterior problems in fluid dynamics.

Abstract

We construct centered rarefaction wave solutions given background solutions to the compressible Euler equations. The flow considered in this article is the homentropic flow of perfect gas governed by compressible Euler equations and the gamma-law equation of state in 3-D spherical symmetry. We prove the existence of local in time rarefactions for general background solutions, which corresponds to one side of the solution to the Riemann problem in spherical symmetry. We also prove the existence of global in time rarefactions for background solutions that are close to constant states with reasonable decay.

Local and Global Results on Three Dimensional Rarefaction Waves in Spherical Symmetry

TL;DR

The paper constructs and analyzes centered rarefaction waves for three-dimensional compressible Euler flow with spherical symmetry, addressing both local existence from general exterior backgrounds and global existence for backgrounds close to constant states with decay. The authors deploy an acoustical-geometry framework, using Riemann invariants and a Cauchy–characteristic approach to propagate data along outgoing null cones, and derive robust energy–flux estimates to control solutions. They establish precise boundary-data propagation, construct local approximate solutions, and prove convergence to genuine rarefaction waves, followed by a global extension under small perturbations of a constant state with decay, including detailed regularity propagation. The results advance understanding of multidimensional rarefaction waves in curved geometries and provide techniques that may inform the full non-symmetric 3D problem and related exterior problems in fluid dynamics.

Abstract

We construct centered rarefaction wave solutions given background solutions to the compressible Euler equations. The flow considered in this article is the homentropic flow of perfect gas governed by compressible Euler equations and the gamma-law equation of state in 3-D spherical symmetry. We prove the existence of local in time rarefactions for general background solutions, which corresponds to one side of the solution to the Riemann problem in spherical symmetry. We also prove the existence of global in time rarefactions for background solutions that are close to constant states with reasonable decay.

Paper Structure

This paper contains 20 sections, 16 theorems, 277 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Review of Shocks and Rarefactions
  3. Problem Setting and Main Results
  4. Review of the Riemann Problem, Riemann invariants
  5. Preparations
  6. Acoustical Metric, Coordinates and Frames
  7. Wave Operator, Energy Estimates
  8. Local Solution for General Background
  9. Determine Data on the Boundary $C_0$
  10. Construct Local Approximate Solutions
  11. Energy Estimates Uniform in $\delta$
  12. Proof of Proposition \ref{['prop:local-base_energy']}
  13. Proof of Proposition \ref{['prop:local-regularity']}
  14. Convergence to the Rarefaction Wave Solution
  15. Global in Time Solution for Background Close to Constant State
  16. ...and 5 more sections

Key Result

Theorem 1.1

Given a smooth spherical symmetric exterior solution $(v, c)$ to the compressible Euler equation, one can construct a piece of (centered) rarefaction wave solution on a fan-shaped region originating from an initial sphere. The solution approaches the 1-D rarefaction solution when we approach the ini

Figures (6)

  • Figure 1: Local and Global
  • Figure 2: 1-D Rarefaction
  • Figure 3: Local Rarefaction
  • Figure 4: Convergence
  • Figure 5: Local Well-Posedness
  • ...and 1 more figures

Theorems & Definitions (26)

  • Theorem 1.1: Local Existence of Rarefaction Wave Solutions
  • Theorem 1.2: Global Existence of Rarefaction Wave Solutions
  • Proposition 2.1: Energy Estimate Framework
  • Theorem 3.1: Local Existence of Rarefaction Solution
  • Remark 3.2
  • Proposition 3.3: Construct Data on $C_0$
  • Proposition 3.4: Local well-posedness of Cauchy-Characteristic Mixed Initial Value Problem
  • Remark 3.5
  • Proposition 3.6: Base Energy Estimates
  • Proposition 3.7: Propagation of Regularity
  • ...and 16 more