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Global well-posedness and large-time behavior of 1D compressible Navier-Stokes equations with density-depending viscosity and vacuum in unbounded domains

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Abstract

We consider the Cauchy problem for one-dimensional (1D) barotropic compressible Navier-Stokes equations with density-depending viscosity and large external forces. Under a general assumption on the density-depending viscosity, we prove that the Cauchy problem admits a unique global strong (classical) solution for the large initial data with vacuum. Moreover, the density is proved to be bounded from above time-independently. As a consequence, we obtain the large time behavior of the solution without external forces.

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References

  1. Amosov A A, Zlotnik A A. Global generalized solutions of the equations of the one-dimensional motion of a viscous heat-conducting gas. Soviet Math Dokl, 1989, 38: 1–5

    MathSciNet  MATH  Google Scholar 

  2. Beirão da Veiga H. Long time behavior for one-dimensional motion of a general barotropic viscous fluid. Arch Ration Mech Anal, 1989, 108: 141–160

    Article  MathSciNet  Google Scholar 

  3. Cho Y, Kim H. On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscripta Math, 2006, 120: 91–129

    Article  MathSciNet  Google Scholar 

  4. Ding S J, Wen H Y, Zhu C J. Global classical large solutions of 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum. J Differential Equations, 2011, 221: 1696–1725

    Article  MathSciNet  Google Scholar 

  5. Feireisl E. Dynamics of Viscous Compressible Fluids. New York: Oxford University Press, 2004

    MATH  Google Scholar 

  6. Hoff D. Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans Amer Math Soc, 1987, 303: 169–181

    MathSciNet  MATH  Google Scholar 

  7. Huang F M, Li J, Xin Z P. Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data. J Math Pures Appl (9), 2006, 86: 471–491

    Article  MathSciNet  Google Scholar 

  8. Huang X D, Li J. Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows. Comm Math Phys, 2013, 324: 147–171

    Article  MathSciNet  Google Scholar 

  9. Huang X D, Li J, Xin Z P. Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations. Comm Pure Appl Math, 2012, 65: 549–585

    Article  MathSciNet  Google Scholar 

  10. Kanel I. A model system of equations for the one-dimensional motion of a gas (in Russian). Differencial’nye Uravnenija, 1968, 4: 721–734

    Google Scholar 

  11. Kazhikhov A V. Cauchy problem for viscous gas equations. Siberian Math J, 1982, 23: 44–49

    Article  Google Scholar 

  12. Li J, Liang Z L. On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum. J Math Pures Appl (9), 2014, 102: 640–671

    Article  MathSciNet  Google Scholar 

  13. Li J, Xin Z P. Some uniform estimates and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows. J Differential Equations, 2006, 221: 275–308

    Article  MathSciNet  Google Scholar 

  14. Li J, Xin Z P. Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum. Ann PDE, 2019, 5: 7

    Article  MathSciNet  Google Scholar 

  15. Li J, Zhang J W, Zhao J N. On the global motion of viscous compressible barotropic flows subject to large external potential forces and vacuum. SIAM J Math Anal, 2015, 47: 1121–1153

    Article  MathSciNet  Google Scholar 

  16. Lions P-L. Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible Models. New York: Oxford University Press, 1998

    MATH  Google Scholar 

  17. Lü B Q, Wang Y X, Wu Y H. On global classical solutions to 1d compressible Navier-Stokes equations with density-dependent viscosity and vacuum. ArXiv:1808.03042, 2018

  18. Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat-conductive gases. J Math Kyoto Univ, 1980, 20: 67–104

    MathSciNet  MATH  Google Scholar 

  19. Serre D. Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible. C R Acad Sci Paris Ser I Math, 1986, 303: 639–642

    MathSciNet  MATH  Google Scholar 

  20. Serre D. On the one-dimensional equation of a viscous, compressible, heat-conducting fluid. C R Acad Sci Paris Ser I Math, 1986, 303: 703–706

    MATH  Google Scholar 

  21. Straškraba I, Zlotnik A. On a decay rate for 1D-viscous compressible barotropic fluid equations. J Evol Equ, 2002, 2: 69–96

    Article  MathSciNet  Google Scholar 

  22. Xin Z P. Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm Pure Appl Math, 1998, 51: 229–240

    Article  MathSciNet  Google Scholar 

  23. Ye Y L. Global classical solution to 1D compressible Navier-Stokes equations with no vacuum at infinity. Math Meth Appl Sci, 2016, 39: 776–795

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by Undergraduate Research Fund of Beijing Normal University (Grant Nos. 2017-150 and 201810027047), and National Natural Science Foundation of China (Grant Nos. 11601218 and 11771382).

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Kexin Li

  2. College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang, 330063, China

    Boqiang Lü

  3. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yixuan Wang

Authors
  1. Kexin Li
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  2. Boqiang Lü
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  3. Yixuan Wang
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Correspondence to Boqiang Lü.

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Li, K., Lü, B. & Wang, Y. Global well-posedness and large-time behavior of 1D compressible Navier-Stokes equations with density-depending viscosity and vacuum in unbounded domains. Sci. China Math. 64, 1231–1244 (2021). https://doi.org/10.1007/s11425-019-9521-4

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  • DOI: https://doi.org/10.1007/s11425-019-9521-4

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