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Time-asymptotics of physical vacuum free boundaries for compressible inviscid flows with damping

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Abstract

In this paper, we prove the leading term of time-asymptotics of the moving vacuum boundary for compressible inviscid flows with damping to be that for Barenblatt self-similar solutions to the corresponding porous media equations obtained by simplifying momentum equations via Darcy’s law plus the possible shift due to the movement of the center of mass, in the one-dimensional and three-dimensional spherically symmetric motions, respectively. This gives a complete description of the large time asymptotic behavior of solutions to the corresponding vacuum free boundary problems. The results obtained in this work are the first ones concerning the large time-asymptotics of physical vacuum boundaries for compressible inviscid fluids, to the best of our knowledge.

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Acknowledgements

This research was supported in part by NSFC Grants 11822107, 12171267 and 11671225. The author would like to thank Professor Chongchun Zeng for helpful discussions.

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  1. Department of Mathematical Sciences & Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Huihui Zeng

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  1. Huihui Zeng
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Correspondence to Huihui Zeng.

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Communicated by A. Malchiodi.

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Zeng, H. Time-asymptotics of physical vacuum free boundaries for compressible inviscid flows with damping. Calc. Var. 61, 59 (2022). https://doi.org/10.1007/s00526-021-02161-9

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