Existence and stability of planar diffusion waves for 2-D Euler equations with damping

Weike Wang Tong Yang

Analysis of PDEs mathscidoc:1912.43970

Journal of Differential Equations, 242, (1), 40-71, 2007.11
To study the non-linear stability of a non-trivial profile for a multi-dimensional systems of gas dynamics, the combination of the Green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential. In this paper, we study the stability of a planar diffusion wave for the isentropic Euler equations with damping in two-dimensional space. By introducing an approximate Green function for the linearized equations around the planar diffusion wave and by applying the energy method, we prove the global existence and the L 2 convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave. The decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the L 2 norm. Furthermore, the constructed approximate Green function in this paper can be
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@inproceedings{weike2007existence,
  title={Existence and stability of planar diffusion waves for 2-D Euler equations with damping},
  author={Weike Wang, and Tong Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210959886589534},
  booktitle={Journal of Differential Equations},
  volume={242},
  number={1},
  pages={40-71},
  year={2007},
}
Weike Wang, and Tong Yang. Existence and stability of planar diffusion waves for 2-D Euler equations with damping. 2007. Vol. 242. In Journal of Differential Equations. pp.40-71. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210959886589534.
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