Structural stability of cylindrical supersonic solutions to the steady Euler-Poisson system

Chunpeng Wang; Zihao Zhang

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arXiv · 2503.15839

Structural stability of cylindrical supersonic solutions to the steady Euler-Poisson system

Chunpeng Wang·Zihao Zhang

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Abstract

This paper concerns the structural stability of smooth cylindrically symmetric supersonic Euler-Poisson flows in nozzles. Both three-dimensional and axisymmetric perturbations are considered. On one hand, we establish the existence and uniqueness of three-dimensional smooth supersonic solutions to the potential flow model of the steady Euler-Poisson system. On the other hand, the existence and uniqueness of smooth supersonic flows with nonzero vorticity to the steady axisymmetric Euler-Poisson system are proved. The problem is reduced to solve a nonlinear boundary value problem for a hyperbolic-elliptic mixed system. One of the key ingredients in the analysis of three-dimensional supersonic irrotational flows is the well-posedness theory for a linear second order hyperbolic-elliptic coupled system, which is achieved by using the multiplier method and the reflection technique to derive the energy estimates. For smooth axisymmetric supersonic flows with nonzero vorticity, the deformation-curl-Poisson decomposition is utilized to reformulate the steady axisymmetric Euler-Poisson system as a deformation-curl-Poisson system together with several transport equations, so that one can design a two-layer iteration scheme to establish the nonlinear structural stability of the background supersonic flow within the class of axisymmetric rotational flows.

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Chunpeng Wang, Zihao Zhang. 2025-03-20. Structural stability of cylindrical supersonic solutions to the steady Euler-Poisson system. https://arxiv.org/abs/2503.15839

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