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Science China Mathematics

, Volume 62, Issue 1, pp 33–62 | Cite as

Convergence to diffusion waves for solutions of Euler equations with time-depending damping on quadrant

  • Haibo Cui
  • Haiyan Yin
  • Changjiang ZhuEmail author
  • Limei Zhu
Articles
  • 29 Downloads

Abstract

This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant (x, t) ∈ ℝ+ × ℝ+,
$${\partial _t}v - {\partial _x}u = 0,{\partial _t}u + {\partial _x}p\left( v \right) = - \frac{\alpha }{{{{\left( {1 + t} \right)}^\lambda }}}u,$$
, with the null-Dirichlet boundary condition or the null-Neumann boundary condition on u. We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends time- asymptotically to the nonlinear diffusion wave. Compared with the previous work about Euler equations with constant coefficient damping, studied by Nishihara and Yang (1999), and Jiang and Zhu (2009, Discrete Contin Dyn Syst), we obtain a general result when the initial perturbation belongs to the same space. In addition, our main novelty lies in the fact that the cut-off points of the convergence rates are different from our previous result about the Cauchy problem. Our proof is based on the classical energy method and the analyses of the nonlinear diffusion wave.

Keywords

Euler equations with time-depending damping nonlinear diffusion waves initial-boundary value problem decay estimates 

MSC(2010)

35L65 76N15 35B45 35B40 

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11331005, 11771150, 11601164 and 11601165).

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Haibo Cui
    • 1
  • Haiyan Yin
    • 1
  • Changjiang Zhu
    • 2
    Email author
  • Limei Zhu
    • 2
  1. 1.School of Mathematical SciencesHuaqiao UniversityQuanzhouChina
  2. 2.School of MathematicsSouth China University of TechnologyGuangzhouChina

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