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Smooth axisymmetric transonic irrotational flows to the steady Euler equations with an external force

Shangkun Weng, Yan Zhou

Abstract

For a class of external forces, we prove the existence and uniqueness of smooth transonic flows to the one dimensional steady Euler system with an external force, which is subsonic at the inlet and flows out at supersonic speed after smoothly accelerating through the sonic point. We then investigate the structural stability of the one-dimensional smooth transonic flows with positive acceleration under axisymmetric perturbations of suitable boundary conditions, and establish the first existence and uniqueness result for smooth axisymmetric transonic irrotational flows. The key point lies on the analysis of a linear second order elliptic-hyperbolic mixed differential equation of Keldysh type with a singular term. Some weighted Sobolev spaces $H_r^m(D) (m=2,3,4)$ are introduced to deal with the singularities near the axis. Compared with the stability analysis in the two dimensional case by Weng and Xin (arXiv:2309.07468), there are several interesting new observations about the structure of the linear mixed type equation with a singular term which play crucial roles in establishing the $H^4_r(D)$ estimate.

Smooth axisymmetric transonic irrotational flows to the steady Euler equations with an external force

Abstract

For a class of external forces, we prove the existence and uniqueness of smooth transonic flows to the one dimensional steady Euler system with an external force, which is subsonic at the inlet and flows out at supersonic speed after smoothly accelerating through the sonic point. We then investigate the structural stability of the one-dimensional smooth transonic flows with positive acceleration under axisymmetric perturbations of suitable boundary conditions, and establish the first existence and uniqueness result for smooth axisymmetric transonic irrotational flows. The key point lies on the analysis of a linear second order elliptic-hyperbolic mixed differential equation of Keldysh type with a singular term. Some weighted Sobolev spaces are introduced to deal with the singularities near the axis. Compared with the stability analysis in the two dimensional case by Weng and Xin (arXiv:2309.07468), there are several interesting new observations about the structure of the linear mixed type equation with a singular term which play crucial roles in establishing the estimate.
Paper Structure (6 sections, 22 theorems, 279 equations)

This paper contains 6 sections, 22 theorems, 279 equations.

Table of Contents

  1. Introduction and main results
  2. Transonic flows to one dimensional steady Euler system with an external force
  3. Smooth axisymmetric transonic irrotational flows
  4. The $H^2_r$ strong solution to the linear second order elliptic-hyperbolic mixed equation
  5. The $H^4_r$ energy estimates and the proof of Theorem \ref{['irro']}.
  6. Appendix

Key Result

Proposition 1.1

( General accelerating transonic flows.) Assume that $(u_0, \rho_0)$ is subsonic and the external force $\bar{f}$ satisfies 1df04 and 1df90. Then there exists a unique accelerating transonic flow $(\bar{u}(x_1), \bar{\rho} (x_1)) \in C ([L_0, L_1])$ which is subsonic in $[L_0, 0)$, supersonic in $(0

Theorems & Definitions (41)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Theorem 1.7
  • ...and 31 more