Acta Mathematica Scientia

, Volume 39, Issue 1, pp 259–283 | Cite as

Stability of Boundary Layer to An Outflow Problem for A Compressible Non-Newtonian Fluid in the Half Space

  • Jie Pan (潘洁)
  • Li Fang (方莉)
  • Zhenhua Guo (郭真华)Email author


This paper investigates the large-time behavior of solutions to an outflow problem for a compressible non-Newtonian fluid in a half space. The main concern is to analyze the phenomena that happens when the compressible non-Newtonian fluid blows out through the boundary. Based on the existence of the stationary solution, it is proved that there exists a boundary layer (i.e., the stationary solution) to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation.

Key words

compressible non-Newtonian fluid stability boundary layer 

2010 MR Subject Classification

35B40 35Q30 76N10 


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  1. [1]
    Il’in A M, Oleĭnik O A. Asymptotic behavior of solutions of the Cauchy problem for some quasi-linear equations for large values of the time (Russian). Mat Sb, 1960, 51(93): 191–216MathSciNetGoogle Scholar
  2. [2]
    Bellout H, Bloom F, Nečas J. Existence, uniqueness and stability of solutions to the initial boundary value problem for bipolar viscous fluids. Differential Integral Equations, 1995, 8(2): 453–464MathSciNetzbMATHGoogle Scholar
  3. [3]
    Fang L, Guo Z H. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure and Applied Analysis, 2017, 16(1): 209–242MathSciNetzbMATHGoogle Scholar
  4. [4]
    Fang L, Li Z L. On the existence of local classical solution for a class of one-dimensional compressible non-Newtonian fluids. Acta Math Sci, 2015, 35B(1): 157–181MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Feireisl E, Liao X, Málek J. Global weak solutions to a class of non-Newtonian compressible fluids. Math Methods Appl Sci, 2015, 38(16): 3482–3494MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Huang F M, Qin X H. Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation. J Differential Equation, 2009, 246(10): 4077–4096MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Kanel’ J I. On a model system of equations of one-dimensional gas motion. Differential Equation, 1968, 4: 374–380zbMATHGoogle Scholar
  8. [8]
    Kawashima S C, Matsusmura A. Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun Math Phys, 1985, 101(1): 97–127MathSciNetCrossRefGoogle Scholar
  9. [9]
    Kawashima S C, Nishibata S, Zhu P. Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space. Commun Math Phys, 2003, 240(3): 483–500MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Liu T P, Smoller J. On the vacuum state for the isentropic gas dynamics equations. Adv Appl Math, 1980, 1(4): 345–359MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Liu T P. Behaviors of solutions for the Burgers equations with boundary corresponding to rarefaction waves. Nonlinear Stud, 2006, 29(29): 293–308MathSciNetGoogle Scholar
  12. [12]
    Málek J, Nečas J, Rokyta M, Ružička M. Weak and Measure-Valued Solution to Evolutionary PDEs. Chapman and Hall, 1996CrossRefzbMATHGoogle Scholar
  13. [13]
    Mamontov A E. Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids. Mathematical Notes, 2000, 68(3): 312–325MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Matsumura A. Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Nonlinear Analysis, 2001, 47(6): 4269–4282MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Matsumura A, Nishihara K J. Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas. Japan J Indust Appl Math, 1986, 3(1): 1–13MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Shi X D. On the stability of rarefaction wave solutions for viscous psystem with boundary effect. Acta Mathematicae Applicatae Sinica, English Series, 2003, 19(2): 341–352MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Shi X D, Wang, T, Zhang Z. Asymptotic stability for one-dimensional motion of non-Newtonian compressible fluids. Acta Mathematicae Applicatae Sinica, English Series, 2014, 30(1): 99–110MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Zhikov V V, Pastukhova S E. On the solvability of the Navier-Stokes system for a compressible non-Newtonian fluid (Russian). Dokl Math, 2009, 73(3): 403–407CrossRefzbMATHGoogle Scholar
  19. [19]
    Yuan H J, Wang C J. Unique solvability for a class of full non-Newtonian fluids of one dimension with vacuum. Z Angew Math Phys, 2009, 60(5): 868–898MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Yuan H J, Yang Z. A class of compressible non-Newtonian fluids with external force and vacuum under no compatibility conditions. Boundary Value Problems, 2016, 2016(1): 201–216MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Xin Z P. On nonlinear stability of contact discontinuities//Hyperbolic Problems: Theory, Numerics, Applications. River Edge, NJ: World Sci Publishing, 1996: 249–256Google Scholar

Copyright information

© Wuhan Institutes of Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  • Jie Pan (潘洁)
    • 1
  • Li Fang (方莉)
    • 1
  • Zhenhua Guo (郭真华)
    • 1
    Email author
  1. 1.Center for Nonlinear Studies, Department of MathematicalNorthwest UniversityXi’anChina

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