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A combination of energy method and spectral analysis for study of equations of gas motion

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Abstract

There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic profiles. In this paper, we will first illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier-Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative systems in fluid dynamics and kinetic models such as the model system of radiating gas and the Vlasov-Poisson-Boltzmann system.

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria

    Renjun Duan

  2. Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Hong Kong, China

    Seiji Ukai

  3. Department of Mathematics, City University of Hong Kong, Hong Kong, China

    Tong Yang

Authors
  1. Renjun Duan
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  2. Seiji Ukai
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  3. Tong Yang
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Correspondence to Renjun Duan.

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Duan, R., Ukai, S. & Yang, T. A combination of energy method and spectral analysis for study of equations of gas motion. Front. Math. China 4, 253–282 (2009). https://doi.org/10.1007/s11464-009-0020-x

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  • DOI: https://doi.org/10.1007/s11464-009-0020-x

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