L2-contraction of large planar shock waves for multi-dimensional scalar viscous conservation laws

Moon-Jin Kang Alexis F Vasseur Yi Wang

Analysis of PDEs mathscidoc:1912.43763

Journal of Differential Equations, 267, (5), 2737-2791, 2019.8
We consider a L 2-contraction (a L 2-type stability) of large viscous shock waves for the multi-dimensional scalar viscous conservation laws, up to a suitable shift by using the relative entropy methods. Quite different from the previous results, we find a new way to determine the shift function, which depends both on the time and space variables and solves a viscous Hamilton-Jacobi type equation with source terms. Moreover, we do not impose any conditions on the anti-derivative variables of the perturbation around the shock profile. More precisely, it is proved that if the initial perturbation around the viscous shock wave is suitably small in L 2-norm, then the L 2-contraction holds true for the viscous shock wave up to a suitable shift function. Note that BV-norm or the L-norm of the initial perturbation and the shock wave strength can be arbitrarily large. Furthermore, as the time t tends to infinity, the L 2-contraction holds
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@inproceedings{moon-jin2019l2-contraction,
  title={L2-contraction of large planar shock waves for multi-dimensional scalar viscous conservation laws},
  author={Moon-Jin Kang, Alexis F Vasseur, and Yi Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205603523641327},
  booktitle={Journal of Differential Equations},
  volume={267},
  number={5},
  pages={2737-2791},
  year={2019},
}
Moon-Jin Kang, Alexis F Vasseur, and Yi Wang. L2-contraction of large planar shock waves for multi-dimensional scalar viscous conservation laws. 2019. Vol. 267. In Journal of Differential Equations. pp.2737-2791. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205603523641327.
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