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Interface behavior and decay rates of compressible Navier-Stokes system with density-dependent viscosity and a vacuum

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Abstract

In this paper, we study the one-dimensional motion of viscous gas near a vacuum, with the gas connecting to a vacuum state with a jump in density. The interface behavior, the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficient μ(ρ) = ρα for any 0 < α < 1; this includes the time-weighted boundedness from below and above. The smoothness of the solution is discussed. Moreover, we construct a class of self-similar classical solutions which exhibit some interesting properties, such as optimal estimates. The present paper extends the results in [Luo T, Xin Z P, Yang T. SIAM J Math Anal, 2000, 31(6): 1175–1191] to the jump boundary conditions case with density-dependent viscosity.

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Authors and Affiliations

  1. School of Mathematics and CNS, Northwest University, Xi’an, 710127, China

    Zhenhua Guo & Xueyao Zhang

  2. School of Mathematics and Information Science, Guangxi University, Nanning, 530004, China

    Zhenhua Guo

Authors
  1. Zhenhua Guo
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  2. Xueyao Zhang
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Correspondence to Xueyao Zhang.

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Conflict of Interest The authors declare no conflict of interest.

Additional information

This work was supported by the NSFC (11931013) and the GXNSF (2022GXNSFDA035078).

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Guo, Z., Zhang, X. Interface behavior and decay rates of compressible Navier-Stokes system with density-dependent viscosity and a vacuum. Acta Math Sci 44, 247–274 (2024). https://doi.org/10.1007/s10473-024-0114-2

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  • DOI: https://doi.org/10.1007/s10473-024-0114-2

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