Skip to main content
SpringerLink
Log in

Zero Dissipation Limit to Rarefaction Waves for the One-Dimensional Compressible Navier-Stokes Equations with Selected Density-Dependent Viscosity

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

This paper is devoted to studying the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations with selected density-dependent viscosity. In particular, we focus our attention on the viscosity taking the form μ(ρ) = ρϵ(ϵ > 0). For the selected density-dependent viscosity, it is proved that the solutions of the one-dimensional compressible Navier-Stokes equations with centered rarefaction wave initial data exist for all time, and converge to the centered rarefaction waves as the viscosity vanishes, uniformly away from the initial discontinuities. New and subtle analysis is developed to overcome difficulties due to the selected density-dependent viscosity to derive energy estimates, in addition to the scaling argument and elementary energy analysis. Moreover, our results extend the studies in [Xin Z P. Comm Pure Appl Math, 1993, 46(5): 621–665].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Jiang S, Xin Z P, Zhang P. Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity. Methods Appl Anal, 2005, 12(3): 239–252

    MathSciNet  MATH  Google Scholar 

  2. Kazhikhov A V, Shelukhin V V. Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J Appl Math Mech, 41 (2): 273–282; translated from Prikl Mat Meh, 1977, 41 (2): 282–291

  3. Lions P L. Mathematical Topics in Fluid Dynamics Vol 2: Compressible Models. Oxford: Oxford Science Publications, 1998

    MATH  Google Scholar 

  4. Liu T P, Xin Z P. Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations. Comm Math Phys, 1988, 118(3): 451–465

    Article  MathSciNet  Google Scholar 

  5. Matsumura A, Nishihara K. Asymptotics toward the rarefaction wave of the solutions of a one-dimensional model system for compressible viscous gas. Japan J Appl Math, 1986, 3(1): 1–13

    Article  MathSciNet  Google Scholar 

  6. Matsumura A, Nishihara K. Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas. Comm Math Phys, 1992, 144(2): 325–335

    Article  MathSciNet  Google Scholar 

  7. Liu T P, Xin Z P, Yang T. Vacuum states for compressible flow. Discrete Contin Dyn Syst, 1998, 4(1): 1–32

    Article  MathSciNet  Google Scholar 

  8. Li H L, Li J, Xin Z P. Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations. Comm Math Phys, 2008, 281: 401–444

    Article  MathSciNet  Google Scholar 

  9. Mellet A, Vasseur A. Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J Math Anal, 2008, 39(4): 1344–1365

    Article  MathSciNet  Google Scholar 

  10. Guo Z H, Jiu Q S, Xin Z P. Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients. SIAM J Math Anal, 2008, 39(5): 1402–1427

    Article  MathSciNet  Google Scholar 

  11. Yang T, Yao Z A, Zhu C J. Compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Comm Partial Differential Equations, 2001, 26(5/6): 965–981

    Article  MathSciNet  Google Scholar 

  12. Yang T, Zhu C J. Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Comm Math Phys, 2002, 230(2): 329–363

    Article  MathSciNet  Google Scholar 

  13. Jiu Q S, Xin Z P. The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients. Kinet Relat Models, 2008, 1(2): 313–330

    Article  MathSciNet  Google Scholar 

  14. Hoff D, Liu T P. The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data. Indiana Univ Math J, 1989, 38(4): 861–915

    Article  MathSciNet  Google Scholar 

  15. Goodman J, Xin Z P. Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch Rat Mech Anal, 1992, 121(3): 235–265

    Article  MathSciNet  Google Scholar 

  16. Xin Z P. Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases. Comm Pure Appl Math, 1993, 46(5): 621–665

    Article  MathSciNet  Google Scholar 

  17. Wang H Y. Viscous limits for piecewise smooth solutions of the p-system. J Math Anal Appl, 2004, 299(2): 411–432

    Article  MathSciNet  Google Scholar 

  18. Chen G Q, Perepelitsa M. Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow. Comm Pure Appl Math, 2010, 63(11): 1469–1504

    Article  MathSciNet  Google Scholar 

  19. Jiang S, Ni G X, Sun W J. Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids. SIAM J Math Anal, 2006, 38(2): 368–384

    Article  MathSciNet  Google Scholar 

  20. Xin Z P, Zeng H H. Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations. J Differential Equations, 2010, 249(4): 827–871

    Article  MathSciNet  Google Scholar 

  21. Wang Y. Zero dissipation limit of the compressible heat-conducting navier-stokes equations in the presence of the shock. Acta Math Sci, 2008, 28B(4): 727–748

    MathSciNet  MATH  Google Scholar 

  22. Ma S X. Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations. J Differential Equations, 2010, 248(1): 95–110

    Article  MathSciNet  Google Scholar 

  23. Huang F M, Wang Y, Yang T. Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinet Relat Models, 2010, 3(4): 685–728

    Article  MathSciNet  Google Scholar 

  24. Huang F M, Jiang S, Wang Y. Zero dissipation limit of full compressible Navier-Stokes equations with a Riemann initial data. Commun Inf Syst, 2013, 13(2): 211–246

    MathSciNet  MATH  Google Scholar 

  25. Huang F M, Wang Y, Yang T. Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to a Riemann problem. Arch Ration Mech Anal, 2012, 203(2): 379–413

    Article  MathSciNet  Google Scholar 

  26. Zhang Y, Pan R, Wang Y, Tan Z. Zero dissipation limit with two interacting shocks of the 1D non-isentropic Navier-Stokes equations. Indiana Univ Math J, 2013, 62(1): 249–309

    Article  MathSciNet  Google Scholar 

  27. Hong H. Zero dissipation limit to contact discontinuity for the compressible Navier-Stokes system of general gas. Acta Math Sci, 2016, 36B(1): 157–172

    Article  MathSciNet  Google Scholar 

  28. Hong H, Wang T. Zero dissipation limit to a Riemann solution for the compressible Navier-Stokes system of general gas. Acta Math Sci, 2017, 37B(5): 1177–1208

    Article  MathSciNet  Google Scholar 

  29. Shi X, Yong Y, Zhang Y. Vanishing viscosity for non-isentropic gas dynamics with interacting shocks. Acta Math Sci, 2016, 36B(6): 1699–1720

    Article  MathSciNet  Google Scholar 

  30. Huang F M, Wang Y, Wang Y, Yang T. The limit of the Boltzmann equation to the Euler equations for Riemann problems. SIAM J Math Anal, 2013, 45(3): 1741–1811

    Article  MathSciNet  Google Scholar 

  31. Hoff D. Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data. Proc Royal Soc Edinburgh A, 1986, 103(3/4): 301–315

    Article  MathSciNet  Google Scholar 

  32. Hoff D, Smoller J. Solutions in the large for certain nonlinear parabolic systems. Ann Inst H Poincaré Anal. Non Linéaire, 1985, 2(3): 213–235

    Article  MathSciNet  Google Scholar 

  33. Jiang S. Global smooth solutions of the equations of a viscous, heat-conducting one-dimensional gas with density-dependent viscosity. Math Nachr, 1998, 190: 169–183

    Article  MathSciNet  Google Scholar 

  34. Feireisl E, Novotný A, Petzeltová H. On the existence of globally defined weak solutions to the Navier-Stokes equations. J Math Fluid Mech, 2001, 3(4): 358–392

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Yifan Su  (苏奕帆) & Zhenhua Guo  (郭真华)

Authors
  1. Yifan Su  (苏奕帆)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhenhua Guo  (郭真华)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhenhua Guo  (郭真华).

Additional information

The second author was supported by the National Natural Science Foundation of China (11671319, 11931013).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Su, Y., Guo, Z. Zero Dissipation Limit to Rarefaction Waves for the One-Dimensional Compressible Navier-Stokes Equations with Selected Density-Dependent Viscosity. Acta Math Sci 41, 1635–1658 (2021). https://doi.org/10.1007/s10473-021-0514-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-021-0514-5

Key words

2010 MR Subject Classification

Search

Navigation