Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

Abstract

This paper concerns the sonic-supersonic structures of the transonic crossflow generated by the steady supersonic flow past an infinite cone of arbitrary cross section. Under the conical assumption, the three-dimensional (3-D) steady Euler equations can be projected onto the unit sphere and the state of fluid can be characterized by the polar and azimuthal angles. Given a segment smooth curve as a conical-sonic line in the polar-azimuthal angle plane, we construct a classical conical-supersonic solution near the curve under some reasonable assumptions. To overcome the difficulty caused by the parabolic degeneracy, we apply the characteristic decomposition technique to transform the Euler equations into a new degenerate hyperbolic system in a partial hodograph plane. The singular terms are isolated from the highly nonlinear complicated system and then can be handled successfully. We establish a smooth local solution to the new system in a suitable weighted metric space and then express the solution in terms of the original variables.

Funding

Natural Science Foundation of Zhejiang Province(LY21A010017); National Natural Science Foundation of China(12171130)