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.
Similar content being viewed by others
References
Bianchini S., Bressan A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. 161(2), 223–342 (2005)
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)
Courant R., Friedrichs K.O.: Supersonic Flow and Shock Waves. Wiley-Interscience, New York (1948)
Gilbarg D.: The existence and limit behavior of the one-dimensional shock layer. Am. J. Math. 73, 256–274 (1951)
Goodman J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95(4), 325–344 (1986)
Goodman J., Xin Z.P.: Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch. Rational Mech. Anal. 121, 235–265 (1992)
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)
Huang F.M., Wang Y., Yang T.: Hydrodynamic limit of the Boltzmann equation with contact discontinuities. Commun. Math. Phys. 295, 293–326 (2010)
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)
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)
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)
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)
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)
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)
Smoller J.: Shock Waves and Reaction-Diffusion Equations. Springer, New York (1994)
Wang H.Y.: Viscous limits for piecewise smooth solutions of the p-system. J. Math. Anal. Appl. 299, 411–432 (2004)
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)
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)
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)
Yu S.H.: Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws. Arch. Rational Mech. Anal. 146, 275–370 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Bressan
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-011-0450-y