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Vanishing Viscosity Limit of the Compressible Navier–Stokes Equations for Solutions to a Riemann Problem

Abstract

We study the vanishing viscosity limit of the compressible Navier–Stokes equations to the Riemann solution of the Euler equations that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible Navier–Stokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the Navier–Stokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.

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References

  1. Bianchini S., Bressan A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. 161(2), 223–342 (2005)

    MathSciNet  MATH  Article  Google Scholar 

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

    MATH  Article  Google Scholar 

  3. Courant R., Friedrichs K.O.: Supersonic Flow and Shock Waves. Wiley-Interscience, New York (1948)

    MATH  Google Scholar 

  4. Gilbarg D.: The existence and limit behavior of the one-dimensional shock layer. Am. J. Math. 73, 256–274 (1951)

    MathSciNet  MATH  Article  Google Scholar 

  5. Goodman J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95(4), 325–344 (1986)

    MathSciNet  ADS  MATH  Article  Google Scholar 

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

    MathSciNet  ADS  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  8. Huang F.M., Wang Y., Yang T.: Hydrodynamic limit of the Boltzmann equation with contact discontinuities. Commun. Math. Phys. 295, 293–326 (2010)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  9. 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 3(4), 685–728 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  10. 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. 38(2), 368–384 (2006)

    MathSciNet  Article  Google Scholar 

  11. Kawashima S., Matsumura A.: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101(1), 97–127 (1985)

    MathSciNet  ADS  MATH  Article  Google Scholar 

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

    MATH  Article  Google Scholar 

  13. Matsumura A., Nishihara K.: On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Jpn. J. Appl. Math. 2(1), 17–25 (1985)

    MathSciNet  MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  15. Smoller J.: Shock Waves and Reaction-Diffusion Equations. Springer, New York (1994)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  17. Wang Y.: Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock. Acta Math. Scientia 28B, 727–748 (2008)

    MATH  Article  Google Scholar 

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

    MATH  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  20. Yu S.H.: Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws. Arch. Rational Mech. Anal. 146, 275–370 (1999)

    ADS  MATH  Article  Google Scholar 

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Correspondence to Tong Yang.

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Communicated by A. Bressan

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Huang, F., Wang, Y. & Yang, T. Vanishing Viscosity Limit of the Compressible Navier–Stokes Equations for Solutions to a Riemann Problem. Arch Rational Mech Anal 203, 379–413 (2012). https://doi.org/10.1007/s00205-011-0450-y

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Keywords

  • Shock Wave
  • Stokes Equation
  • Euler Equation
  • Travel Wave Solution
  • Rarefaction Wave