Geometric Structures of Pseudo-Sonic Curves in Self-Similar Solutions of the Euler Equations for Potential Flow

Technical Summary: Geometric Structures of Pseudo-Sonic Curves in Self-Similar Solutions of the Euler Equations for Potential Flow

Problem Statement
This paper investigates the geometric structures of pseudo-sonic curves within two-dimensional self-similar solutions of the Euler equations for potential flow. The study specifically addresses scenarios involving non-uniform supersonic incoming states, a significant generalization over previous work which largely focused on uniform incoming flows. Mathematically, the governing equation is a second-order nonlinear partial differential equation of mixed hyperbolic-elliptic type, where the type degeneracy occurs precisely along the pseudo-sonic curve (the locus where the pseudo-velocity magnitude equals the local sonic speed). The central challenge is to characterize the geometry of this curve and the regularity of the solution across it without assuming a priori that the curve possesses a specific shape (e.g., circular).

Methodology
The authors develop rigorous analytical approaches to characterize the pseudo-sonic curve, denoted as $\Gamma_{sonic}$. The methodology proceeds through several distinct stages:

1. Classification of Sonic Points: The authors introduce the concept of "exceptional" points on the pseudo-sonic curve. A point is defined as exceptional if the pseudo-velocity vector is normal to the curve at that point.
2. Case Analysis based on Velocity Orientation:
   • Normal Velocity Case: The authors prove that if the pseudo-velocity is normal to the curve at every point, the curve must be a circular arc.
   • Tangential Velocity Case: They analyze the scenario where the pseudo-velocity is tangential to the curve. Depending on the normal derivative of the sonic speed ($\partial_n c$), they demonstrate that the curve must either be a straight line or strictly convex when viewed from the pseudo-supersonic region.
3. Streamline Structure: The geometric properties of streamlines in the neighborhood of the pseudo-sonic curve are analyzed, establishing conditions under which these streamlines are convex or possess zero curvature at the sonic point.
4. Regularity and Contradiction Arguments: To establish that the velocity must be normal to the curve in general settings, the authors introduce a new auxiliary function $\psi_Q$ (the difference between the potential and a reference potential with constant velocity). By analyzing the second-order derivatives of $\psi_Q$ from both sides of the sonic curve and employing barrier functions and maximum principles, they derive sufficient conditions under which the pseudo-velocity must be normal to the curve. This involves proving that if the velocity were not normal, it would lead to a contradiction regarding the regularity or behavior of the solution near the curve.

Key Contributions and Results
The paper establishes several theorems that provide a precise characterization of the pseudo-sonic curve's geometry:

• Theorem 2.1: Proves that if all pseudo-sonic points are exceptional (velocity is normal), the pseudo-sonic curve lies on a circle of constant radius, and the sonic speed is constant along the curve.
• Theorems 2.2–2.3: Characterize the geometry when the velocity is tangential. If the normal derivative of the sonic speed is zero, the curve is a straight line. If the derivative is non-zero, the curve is strictly convex from the supersonic side.
• Theorems 2.5–2.6: Provide sufficient conditions ensuring that a pseudo-sonic point is exceptional (velocity is normal). These theorems rely on natural assumptions regarding the local behavior of the solution (specifically, bounds on the function $\psi_Q$). Crucially, these results do not assume a priori that the velocity is normal or tangential; rather, they deduce the normality from the solution's local estimates.
• Regularity Results: The paper proves that under the condition that the pseudo-velocity is not normal and $\partial_\perp c = 0$, the solution is $C^2$ across the sonic curve. Conversely, if the solution is a small perturbation of a uniform flow solution, the sonic points are shown to be exceptional.
• Application to Shock Reflection-Diffraction: Applying the developed theory to the shock reflection-diffraction problem with non-uniform incoming flow, the authors prove that if the solution is a $C^2$-small perturbation (in either the pseudo-supersonic or pseudo-subsonic region) of the solution constructed for uniform incoming flow (as in [9]), the pseudo-sonic curve must be an arc of a circle. Furthermore, on this arc, the density and velocity are constant, corresponding to the radius and center of the arc, respectively. The solution is shown to be $C^{2,\alpha}$ regular in the pseudo-subsonic region up to the sonic arc (excluding point $P_1$).

Significance and Claims
The authors claim that this work provides the first rigorous result describing the shape of the pseudo-sonic curve and the behavior of solutions near it without assuming a priori that the curve is an arc. While previous works (e.g., [1, 2, 5–7, 9, 11, 16, 17]) established that sonic arcs are circular for uniform incoming flows, the geometry for non-uniform states was unclear, with the possibility of the circular structure breaking down.

The paper demonstrates that for the important case where the solution is a small perturbation of a uniform incoming state, the circular structure is preserved. The techniques and ideas developed, particularly the use of the auxiliary function $\psi_Q$ and the analysis of mixed-type degeneracies, are presented as applicable to other nonlinear problems involving similar degeneracies. The results offer a precise characterization of the geometry of pseudo-sonic curves and the regularity of transonic flows in self-similar settings.