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On Global Multidimensional Supersonic Flows with Vacuum States at Infinity

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Abstract

In this paper, we are concerned with the global existence and stability of a smooth supersonic flow with vacuum state at infinity in a three-dimensional infinitely long divergent nozzle. The flow is described by a three-dimensional steady potential equation, which is multi-dimensional quasilinear hyperbolic (but degenerate at infinity) with respect to the supersonic direction, and whose linearized part admits the form \({{\partial_t^2-\frac{1}{(1+t)^{2(\gamma-1)}}(\partial_1^2+\partial_2^2)+\frac{2(\gamma-1)}{1+t}\partial_t}}\) for \({{1 < \gamma < 2}}\). From the physical point of view, due to the expansive geometric property of the divergent nozzle and the mass conservation of gases, the moving gases in the nozzle will gradually become rarefactive and tend to vacuum states at infinity, which implies that such a smooth supersonic flow should be globally stable for small perturbations since there are no strong resulting compressions in the motion of the flow. We will confirm such a global stability phenomenon by rigorous mathematical proofs and further show that there do not exist vacuum domains in any finite parts of the nozzle.

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Authors and Affiliations

  1. Faculty of Science, Jiangsu University, Zhenjiang, 212013, Jiangsu, China

    Xu Gang

  2. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Yin Huicheng

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  1. Xu Gang
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  2. Yin Huicheng
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Correspondence to Yin Huicheng.

Additional information

Communicated by F. Lin

X. Gang was supported by the National Natural Science Foundation of China (No. 11101190) and Natural Science Fundamental Research Project of Jiangsu Colleges (No. 10KLB110002).

Y. Huicheng was supported by the NSFC (No. 11025105) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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Gang, X., Huicheng, Y. On Global Multidimensional Supersonic Flows with Vacuum States at Infinity. Arch Rational Mech Anal 218, 1189–1238 (2015). https://doi.org/10.1007/s00205-015-0878-6

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