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Structural stability of three dimensional transonic shock flows with an external force

Shangkun Weng, Zihao Zhang, Yan Zhou

Abstract

We establish the existence and uniqueness of the transonic shock solution for steady isentropic Euler system with an external force in a rectangular cylinder under the three-dimensional perturbations for the incoming supersonic flow, the exit pressure and the external force. The external force has a stabilization effect on the transonic shocks in flat nozzles and the transonic shock is completely free, we do not require it passing through a fixed point. By utilizing the deformation-curl decomposition to decouple the hyperbolic and elliptic modes in the steady Euler system effectively and reformulating the Rankine-Hugoniot conditions, the transonic shock problem is reduced to a deformation-curl first order system for the velocity field with nonlocal terms supplementing with an unusual second order differential boundary condition on the shock front, an algebraic equation for determining the shock front and two transport equations for the Bernoulli's quantity and the first component of the vorticity.

Structural stability of three dimensional transonic shock flows with an external force

Abstract

We establish the existence and uniqueness of the transonic shock solution for steady isentropic Euler system with an external force in a rectangular cylinder under the three-dimensional perturbations for the incoming supersonic flow, the exit pressure and the external force. The external force has a stabilization effect on the transonic shocks in flat nozzles and the transonic shock is completely free, we do not require it passing through a fixed point. By utilizing the deformation-curl decomposition to decouple the hyperbolic and elliptic modes in the steady Euler system effectively and reformulating the Rankine-Hugoniot conditions, the transonic shock problem is reduced to a deformation-curl first order system for the velocity field with nonlocal terms supplementing with an unusual second order differential boundary condition on the shock front, an algebraic equation for determining the shock front and two transport equations for the Bernoulli's quantity and the first component of the vorticity.
Paper Structure (7 sections, 8 theorems, 213 equations)

This paper contains 7 sections, 8 theorems, 213 equations.

Table of Contents

  1. Introduction and main results
  2. The reformulation of the transonic shock problem
  3. The deformation-curl decomposition to the steady Euler system
  4. The reformulation of the Rankine-Hugoniot conditions and boundary conditions
  5. Fix the domain and the reformulation of the problem
  6. Iteration scheme and the proof of Theorem \ref{['thm1']}
  7. Appendix

Key Result

Lemma 1.1

Suppose that the initial state $(u_0,\rho_0)$ at $x_1=L_0$ is supersonic and the external force $\bar{f}>0$ for any $x_1 \in [L_0, L_1]$, there are two positive constants $P_0,P_1$ such that if the end pressure $P_e\in (P_1, P_0)$, there exists a unique piecewise transonic shock solution with a shock located at $x_1=L_{s}\in (L_0,L_1)$. Across the shock, the Rankine-Hugoniot conditions and entrop

Theorems & Definitions (10)

  • Lemma 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof