Geometric Structures of Pseudo-Sonic Curves in Self-Similar Solutions of the Euler Equations for Potential Flow
We are concerned with the geometric structures of pseudo-sonic curves in two-dimensional self-similar solutions for the Euler equations for potential flow, allowing for non-uniform supersonic states. Mathematically, the governing second-order potential flow equation is of mixed hyperbolic-elliptic type, with degeneracy occurring along the pseudo-sonic curve. In this paper, we develop rigorous analytical approaches to analyze the geometric structures of pseudo-sonic curves in such self-similar solutions. We first show that the pseudo-sonic curve is necessarily a circle if the pseudo-velocity at each point is a normal to the curve. We then analyze the general case in which the pseudo-velocity on the pseudo-sonic point is not a normal to the curve, and study the geometric properties of streamlines in a neighborhood of the pseudo-sonic curve. Next, we establish two theorems that provide sufficient conditions ensuring that the pseudo-velocity at a pseudo-sonic point is normal to the curve, under natural assumptions on the local behavior of the solution. These results yield a precise characterization of the geometry of pseudo-sonic curves. Finally, we apply the developed theory to the shock reflection-diffraction problem with non-uniform incoming flow. We prove that the pseudo-sonic curve must be an arc if the solution is a $C^2$-small perturbation, either in the pseudo-supersonic or pseudo-subsonic region, of a solution with uniform incoming flow. In particular, the density and velocity must be constant, corresponding to the radius and the center of the pseudo-sonic arc, respectively. Moreover, we prove that the solution is $C^{2,\alpha}$-regular in the pseudo-subsonic region up to the sonic arc (except at point $P_1$). The techniques and ideas developed in this paper are expected to be applicable to other nonlinear problems involving similar mixed-type degeneracies.