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arXiv · 2508.16904

Uniqueness of transonic shock solutions in general approximate nozzles for steady potential flow

Minghong Han·Bingsong Long·Hairong Yuan
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Abstract

We study the uniqueness of solutions with a transonic shock in a two-dimensional Riemannian manifold with a special metric, which can be regarded as an approximate model of the general physical nozzles, within a class of transonic shock solutions for steady potential flow. We first prove the uniqueness of these solutions on the unit 2-sphere: for given uniform supersonic upstream flow at the entry, there exists a unique uniform pressure at the exit such that a transonic shock solution exists in the sphere, which is unique modulo a translation. A similar result is then extended to a class of manifolds. Mathematically, it is equivalent to showing a uniqueness theorem for a free boundary problem of a second-order elliptic-hyperbolic mixed-type partial differential equation in the general approximate nozzles. The proof is based on the maximum/comparison principle with a suitable special transonic shock solution as a comparison function.

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Minghong Han, Bingsong Long, Hairong Yuan. 2025-08-29. Uniqueness of transonic shock solutions in general approximate nozzles for steady potential flow. https://arxiv.org/abs/2508.16904

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