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Existence and stability of cylindrical transonic shock solutions under three dimensional perturbations

Shangkun Weng, Zhouping Xin

Abstract

We establish the existence and stability of cylindrical transonic shock solutions under three dimensional perturbations of the incoming flows and the exit pressure without any further restrictions on the background transonic shock solutions. The strength and position of the perturbed transonic shock are completely free and uniquely determined by the incoming flows and the exit pressure. The optimal regularity is obtained for all physical quantities, and the velocity, the Bernoulli's quantity, the entropy and the pressure share the same regularity. The approach is based on the deformation-curl decomposition to the steady Euler system introduced by the authors to decouple the hyperbolic and elliptic modes effectively. However, one of the key elements in application of the deformation-curl decomposition is to find a decomposition of the Rankine-Hugoniot conditions, which shows the mechanism of determining the shock front uniquely by an algebraic equation and also gives an unusual second order differential boundary conditions on the shock front for the first order deformation-curl system. After homogenizing the curl system and introducing a potential function, this unusual condition on the shock front becomes the Poisson equation with homogeneous Neumann boundary condition on the intersection of the shock front and the cylinder walls from which an oblique boundary condition for the potential function can be uniquely derived.

Existence and stability of cylindrical transonic shock solutions under three dimensional perturbations

Abstract

We establish the existence and stability of cylindrical transonic shock solutions under three dimensional perturbations of the incoming flows and the exit pressure without any further restrictions on the background transonic shock solutions. The strength and position of the perturbed transonic shock are completely free and uniquely determined by the incoming flows and the exit pressure. The optimal regularity is obtained for all physical quantities, and the velocity, the Bernoulli's quantity, the entropy and the pressure share the same regularity. The approach is based on the deformation-curl decomposition to the steady Euler system introduced by the authors to decouple the hyperbolic and elliptic modes effectively. However, one of the key elements in application of the deformation-curl decomposition is to find a decomposition of the Rankine-Hugoniot conditions, which shows the mechanism of determining the shock front uniquely by an algebraic equation and also gives an unusual second order differential boundary conditions on the shock front for the first order deformation-curl system. After homogenizing the curl system and introducing a potential function, this unusual condition on the shock front becomes the Poisson equation with homogeneous Neumann boundary condition on the intersection of the shock front and the cylinder walls from which an oblique boundary condition for the potential function can be uniquely derived.
Paper Structure (7 sections, 9 theorems, 246 equations)

This paper contains 7 sections, 9 theorems, 246 equations.

Table of Contents

  1. Introduction and main results
  2. The reformulation of the transonic shock problem
  3. The deformation-curl decomposition of the steady Euler equations
  4. The reformulation of the Rankine-Hugoniot conditions and boundary conditions
  5. The coordinate transformation
  6. Proof of Theorem \ref{['main']}
  7. Appendix: The compatibility conditions

Key Result

Lemma 1.1

For the incoming data given in super1 satisfying the compatibility conditions then there exists $\epsilon_0>0$ depending only on the background solution and the boundary data, such that for any $0<\epsilon<\epsilon_0$, there exists a unique $C^{2,\alpha}(\overline{\Omega})$ solution $\bm\Psi^-=(U_1^-,U_2^-,U_3^-,P^-,K^-)(r,\theta,x_3)$ to euler-cyl with super1 and slip1, whic and

Theorems & Definitions (24)

  • Lemma 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 14 more