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Blowup for Projected \(\mathbf {2}\)-Dimensional Rotational \({\mathbf {C}}^{2}\) Solutions of Compressible Euler Equations

  • Manwai YuenEmail author
Article

Abstract

The compressible Euler equations are the classical model in fluid dynamics. In this study, we investigate the life span of the projected 2-dimensional rotational \(C^{2}\) non-vacuum solutions of the Euler equations. By examining the corresponding projected 2-dimensional solutions,
$$\begin{aligned} (\rho (t,x_{1},x_{2}),u_{1}(t,x_{1},x_{2}),u_{2}(t,x_{1},x_{2}),0), \end{aligned}$$
in \(\mathbf {R}^{3}\), we prove that there exist the corresponding blowup results for the rotational \(C^{2}\) solutions with a sufficiently large initial functional
$$\begin{aligned} H(0)= {\displaystyle \int _{\mathbf {R}^{3}}} \vec {x}\cdot \vec {u}_{0}dV. \end{aligned}$$

Keywords

Compressible Euler equations Blowup Rotational solutions Initial value problem Functional method Non-vacuum 

Mathematics Subject Classification

35B44 35L67 35Q31 76U05 35B30 

Notes

Acknowledgements

The author thanks for the anonymous reviewers’ valuable comments for improving the quality of this article. This research was partially supported by the Dean’s Research Fund 2015-16 (FLASS/DRF/SFRS-6) from the Education University of Hong Kong.

Compliance with ethical standards

Conflict of interest

The author declare that there is no conflict of interest.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Information TechnologyThe Education University of Hong KongTai PoHong Kong

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