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Nonlinear stability of rarefaction waves for a viscous radiative and reactive gas with large initial perturbation

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Abstract

We investigate the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional compressible Navier-Stokes type system for a viscous, compressible, radiative and reactive gas, where the constitutive relations for the pressure p, the specific internal energy e, the specific volume v, the absolute temperature θ, and the specific entropy s are given by p = Rθ/v + aθ4/3, e = Cvθ + avθ4, and s = Cv ln θ + 4avθ3/3 + Rln v with R > 0, Cv > 0 and a > 0 being the perfect gas constant, the specific heat and the radiation constant, respectively. For such a specific gas motion, a somewhat surprising fact is that, generally speaking, the pressure \(\widetilde{p}(v,s)\) is not a convex function of the specific volume v and the specific entropy s. Even so, we show in this paper that the rarefaction waves are time-asymptotically stable for large initial perturbation provided that the radiation constant a and the strength of the rarefaction waves are sufficiently small. The key point in our analysis is to deduce the positive lower and upper bounds on the specific volume and the absolute temperature, which are uniform with respect to the space and the time variables, but are independent of the radiation constant a.

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Acknowledgements

The first author and the second author were supported by the Fundamental Research Funds for the Central Universities and National Natural Science Foundation of China (Grant Nos. 11731008 and 11671309). The second author was supported by the Fundamental Research Funds for the Central Universities (Grant No. YJ201962). The third author was supported by National Postdoctoral Program for Innovative Talents of China (Grant No. BX20180054). The authors express their thanks to the anonymous referees for their helpful comments and suggestions, which led to the improvement of the presentation of the paper. Last but not least, the authors thank Professor Huijiang Zhao for his support and encouragement.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Guiqiong Gong

  2. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China

    Guiqiong Gong

  3. College of Mathematics, Sichuan University, Chengdu, 610064, China

    Lin He

  4. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Yongkai Liao

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  1. Guiqiong Gong
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  2. Lin He
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Correspondence to Yongkai Liao.

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Gong, G., He, L. & Liao, Y. Nonlinear stability of rarefaction waves for a viscous radiative and reactive gas with large initial perturbation. Sci. China Math. 64, 2637–2666 (2021). https://doi.org/10.1007/s11425-020-1686-6

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