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Global solutions for supersonic flow of a Chaplygin gas past a conical wing with a shock wave detached from the leading edges

Bingsong Long

TL;DR

The paper studies global solutions to the 3D steady Euler equations for a Chaplygin gas flowing past a diamond-shaped conical wing with a shock detached from the leading edges. It leverages the linear degeneracy of the Chaplygin gas to reformulate the problem as an oblique-derivative boundary-value problem for a nonlinear degenerate elliptic equation in conical coordinates, and proves existence via a vanishing-viscosity regularization and a continuity method. A key contribution is the Lipschitz estimate that yields local uniform ellipticity away from the degenerate boundary and enables a rigorous existence theory for the reformulated problem, including careful treatment of the boundary conditions and corners. The results establish global solvability for cone angles $\sigma_1,\sigma_2$ below a critical limit $\sigma_{\infty}$, contributing to the mathematical understanding of shock detachment in Chaplygin-gas flows and the associated degenerate elliptic PDE framework.

Abstract

In this paper, we first investigate the mathematical aspects of supersonic flow of a Chaplygin gas past a conical wing with diamond-shaped cross sections in the case of a shock wave detached from the leading edges. The flow under consideration is governed by the three-dimensional steady compressible Euler equations. For the Chaplygin gas, all characteristics are linearly degenerate, and shocks are reversible and characteristic. Using these properties, we can determine the location of the shock in advance and reformulate our problem as an oblique derivative problem for a nonlinear degenerate elliptic equation in conical coordinates. By establishing a Lipschitz estimate, we show that the equation is uniformly elliptic in any subdomain strictly away from the degenerate boundary, and then further prove the existence of a solution to the problem via the continuity method and vanishing viscosity method.

Global solutions for supersonic flow of a Chaplygin gas past a conical wing with a shock wave detached from the leading edges

TL;DR

The paper studies global solutions to the 3D steady Euler equations for a Chaplygin gas flowing past a diamond-shaped conical wing with a shock detached from the leading edges. It leverages the linear degeneracy of the Chaplygin gas to reformulate the problem as an oblique-derivative boundary-value problem for a nonlinear degenerate elliptic equation in conical coordinates, and proves existence via a vanishing-viscosity regularization and a continuity method. A key contribution is the Lipschitz estimate that yields local uniform ellipticity away from the degenerate boundary and enables a rigorous existence theory for the reformulated problem, including careful treatment of the boundary conditions and corners. The results establish global solvability for cone angles below a critical limit , contributing to the mathematical understanding of shock detachment in Chaplygin-gas flows and the associated degenerate elliptic PDE framework.

Abstract

In this paper, we first investigate the mathematical aspects of supersonic flow of a Chaplygin gas past a conical wing with diamond-shaped cross sections in the case of a shock wave detached from the leading edges. The flow under consideration is governed by the three-dimensional steady compressible Euler equations. For the Chaplygin gas, all characteristics are linearly degenerate, and shocks are reversible and characteristic. Using these properties, we can determine the location of the shock in advance and reformulate our problem as an oblique derivative problem for a nonlinear degenerate elliptic equation in conical coordinates. By establishing a Lipschitz estimate, we show that the equation is uniformly elliptic in any subdomain strictly away from the degenerate boundary, and then further prove the existence of a solution to the problem via the continuity method and vanishing viscosity method.

Paper Structure

This paper contains 9 sections, 6 theorems, 84 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The location of the shock
  4. BVP for a nonlinear degenerate elliptic equation
  5. The Proof of Theorem \ref{['thm4']}
  6. The strategy of the proof
  7. Lipschitz estimate
  8. The Proof of Theorem \ref{['thm4']}
  9. The ellipticity of equation (2-D eq.)

Key Result

Theorem 1.1

Suppose that the conical wing $\mathcal{C}_{\sigma_1}^{\sigma_2}$ is defined by eq:conical body, and the state $(\rho_{\infty},\bm{v}_{\infty}=(0,0,v_{3\infty}))$ of the oncoming flow is uniform and supersonic. Then there exists a critical angle $\sigma_{\infty}=\sigma_{\infty}(\rho_{\infty},\bm{v}_

Figures (3)

  • Figure 1: The generic hypersonic configurations.
  • Figure 2: Supersonic flow past a conical wing with diamond-shaped cross sections.
  • Figure 3: Domain in the conical coordinates.

Theorems & Definitions (14)

  • Theorem 1.1: Main Theorem
  • Theorem 2.1
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Lemma 3.3
  • proof
  • ...and 4 more