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Some Uniform Estimates and Large-Time Behavior of Solutions to One-Dimensional Compressible Navier–Stokes System in Unbounded Domains with Large Data

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Abstract

We study the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier–Stokes system describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. The temperature is proved to be bounded from below and above, independent of both time and space. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity. Note that the initial data can be arbitrarily large. This result is proved by using elementary energy methods.

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Authors and Affiliations

  1. Institute of Applied Mathematics, AMSS & Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Jing Li

  2. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, People’s Republic of China

    Zhilei Liang

Authors
  1. Jing Li
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  2. Zhilei Liang
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Correspondence to Jing Li.

Additional information

Communicated by T.-P. Liu

This work was partially supported by the National Center for Mathematics and Interdisciplinary Sciences, CAS, and NNSFC 11371348, 11525106, 11226163, and 11301422.

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Li, J., Liang, Z. Some Uniform Estimates and Large-Time Behavior of Solutions to One-Dimensional Compressible Navier–Stokes System in Unbounded Domains with Large Data. Arch Rational Mech Anal 220, 1195–1208 (2016). https://doi.org/10.1007/s00205-015-0952-0

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