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The Morawetz Problem for Supersonic Flow with Cavitation

Gui-Qiang G. Chen, Tristan P. Giron, Simon M. Schulz

Abstract

We are concerned with the existence and compactness of entropy solutions of the compressible Euler system for two-dimensional steady potential flow around an obstacle for a polytropic gas with supersonic far-field velocity. The existence problem, initially posed by Morawetz \cite{morawetz85} in 1985, has remained open since then. In this paper, we establish the first complete existence theorem for the Morawetz problem by developing a new entropy analysis, coupled with a vanishing viscosity method and compensated compactness ideas. The main challenge arises when the flow approaches cavitation, leading to a loss of strict hyperbolicity of the system and a singularity of the entropy equation, particularly for the case of adiabatic exponent $γ=3$. Our analysis provides a complete description of the entropy and entropy-flux pairs via the Loewner--Morawetz relations, which, in turn, leads to the establishment of a compensated compactness framework. As direct applications of our entropy analysis and the compensated compactness framework, we obtain the compactness of entropy solutions and the weak continuity of the compressible Euler system in the supersonic regime.

The Morawetz Problem for Supersonic Flow with Cavitation

Abstract

We are concerned with the existence and compactness of entropy solutions of the compressible Euler system for two-dimensional steady potential flow around an obstacle for a polytropic gas with supersonic far-field velocity. The existence problem, initially posed by Morawetz \cite{morawetz85} in 1985, has remained open since then. In this paper, we establish the first complete existence theorem for the Morawetz problem by developing a new entropy analysis, coupled with a vanishing viscosity method and compensated compactness ideas. The main challenge arises when the flow approaches cavitation, leading to a loss of strict hyperbolicity of the system and a singularity of the entropy equation, particularly for the case of adiabatic exponent . Our analysis provides a complete description of the entropy and entropy-flux pairs via the Loewner--Morawetz relations, which, in turn, leads to the establishment of a compensated compactness framework. As direct applications of our entropy analysis and the compensated compactness framework, we obtain the compactness of entropy solutions and the weak continuity of the compressible Euler system in the supersonic regime.
Paper Structure (38 sections, 47 theorems, 347 equations, 2 figures)

This paper contains 38 sections, 47 theorems, 347 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Morawetz Problem and Main Theorems
  3. Problem setup
  4. Main theorems
  5. Connections with the isentropic Euler equations
  6. Hyperbolicity and Entropy Structure
  7. Hyperbolicity and genuine nonlinearity
  8. Riemann invariants
  9. Entropy structure
  10. Entropy Generator I: Regular Kernel
  11. Computing the regular kernel coefficients
  12. Asymptotic analysis of the regular kernel
  13. Properties of the characteristic speed
  14. Coefficients of the regular kernel in the vicinity of the vacuum
  15. Analysis of the remainder term for the regular kernel
  16. ...and 23 more sections

Key Result

Theorem 2.1

There exist distributional solutions $H^\mathbf{r}$ and $H^\mathbf{s}$ with supported in $\mathcal{K}$ of the initial value problems in $(0,\nu_{\mathrm{cr}}) \times \mathbb{R}$ : for which $H^\mathbf{r}$ and $H^\mathbf{s}$ are respectively called the regular kernel and the singular kernel of eq:ent gen intro.

Figures (2)

  • Figure 2.1: The Morawetz problem for two-dimensional steady supersonic flow past an obstacle $O$.
  • Figure 8.1: The posed domain $\mathscr{D}$ for the viscous approximate problems.

Theorems & Definitions (84)

  • Theorem 2.1
  • Remark 2.1: Generating Entropy Pairs
  • Definition 2.2: Entropy Solutions
  • Theorem 2.2: Global Existence of Entropy Solutions
  • Theorem 2.3: Compensated Compactness Framework
  • Theorem 2.4: Compactness and Weak Continuity
  • Proposition 3.1: Hyperbolicity and Genuine Nonlinearity
  • Proposition 3.2: Riemann Invariants
  • proof : Proof of Proposition \ref{['lem:hyperbolic and gn']}
  • proof : Proof of Proposition \ref{['lem:riemann invariants']}
  • ...and 74 more