Abstract
By introducing a new averaged quantity with a fast decay weight to perform Sideris’s argument (Commun Math Phys 101:475–485, 1985) developed for the Euler equations, we extend the formation of singularities of classical solution to the 3D Euler equations established in Makino et al. (Jpn J Appl Math 3:249–257, 1986) and Sideris (1985) for the initial data with compactly supported disturbances to the spherically symmetric solution with general initial data in Sobolev space. Moreover, we also prove the formation of singularities of the spherically symmetric solutions to the 3D Euler–Poisson equations, but remove the compact support assumptions on the initial data in Makino and Perthame (Jpn J Appl Math 7:165–170, 1990) and Perthame (Jpn J Appl Math 7:363–367, 1990). Our proof also simplifies that of Lei et al. (Math Res Lett 20:41–50, 2013) for the Euler equations and is undifferentiated in dimensions.
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Acknowledgements
The research of Li was supported partially by the National Natural Science Foundation of China (No. 11461161007 and 11671384), and the “Capacity Building for Sci-Tech Innovation - Fundamental Scientific Research Funds 007175304800 and 025185305000/182.” The research of Wang was supported by Grant Nos. 231668 and 250070 from the Research Council of Norway.
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Li, HL., Wang, Y. Formation of singularities of spherically symmetric solutions to the 3D compressible Euler equations and Euler–Poisson equations. Nonlinear Differ. Equ. Appl. 25, 39 (2018). https://doi.org/10.1007/s00030-018-0534-6
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DOI: https://doi.org/10.1007/s00030-018-0534-6