Global existence of smooth solutions and convergence to Barenblatt solutions for the physical vacuum free boundary problem of compressible Euler equations with damping

TAO LUO Georgetown University Huihui Zeng Tsinghua University

Analysis of PDEs mathscidoc:1703.03011

Distinguished Paper Award in 2017

Comm. Pure Appl. Math. , 69
For the physical vacuum free boundary problem with the sound speed being C^1/2-Holder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates, and elliptic estimates.
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@inproceedings{taoglobal,
  title={Global existence of smooth solutions and convergence to Barenblatt solutions for the physical vacuum free boundary problem of compressible Euler equations with damping},
  author={TAO LUO, and Huihui Zeng},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170329201742063337734},
  booktitle={Comm. Pure Appl. Math. },
  volume={69},
}
TAO LUO, and Huihui Zeng. Global existence of smooth solutions and convergence to Barenblatt solutions for the physical vacuum free boundary problem of compressible Euler equations with damping. Vol. 69. In Comm. Pure Appl. Math. . http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170329201742063337734.
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