Abstract
Both the global well-posedness for large data and the vanishing shear viscosity limit with a boundary layer to the compressible Navier–Stokes system with cylindrical symmetry are studied under a general condition on the heat conductivity coefficient that, in particular, includes the constant coefficient. The thickness of the boundary layer is proved to be almost optimal. Moreover, the optimal L 1 convergence rate in terms of shear viscosity is obtained for the angular and axial velocity components.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chapman, S., Cowling, T.G.: The mathematical theory of nonuniform gases. In: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge Math. Lib., 3rd edn. Cambridge University Press, Cambridge, (1990)
Fan J., Jiang S., Nakamura G.: Vanishing shear viscosity limit in the magnetohydrodynamic equations. Commun. Math. Phys. 270, 691–708 (2007)
Fan J., Jiang S.: Zero shear viscosity limit of the Navier–Stokes equations of compressible isentropic fluids with cylindrical symmetry. Rend. Sem. Mat. Univ. Politec. Torino 65, 35–52 (2007)
Frid H., Shelukhin V.V.: Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry. SIAM J. Math. Anal. 31, 1144–1156 (2000)
Frid H., Shelukhin V.V.: Boundary layer for the Navier–Stokes equations of compressible fluids. Commun. Math. Phys. 208, 309–330 (1999)
Hoff D., Jessen H.K.: Symmetric nonbarotropic flows with large data and forces. Arch. Ration. Mech. Anal. 173, 297–343 (2004)
Jessen H.K., Karper T.K.: One-dimensional compressible flow with temperature dependent transport coefficients. SIAM J. Math. Anal. 42, 904–930 (2010)
Jiang S., Zhang J.W.: Boundary layers for the Navier–Stokes equations of compressible heat-conducting flows with cylindrical symmetry. SIAM J. Math. Anal. 41, 237–268 (2009)
Kazhikhov A., Shelukhin V.V.: Unique global solutions in time of initial boundary value problems for one-dimensional equations of a viscous gas. PMMJ Appl. Math. Mech. 41, 273–283 (1977)
Landau L.D., Lifshitz E.M.: Fluid Mechanics, 2nd edn. Pergamon Press Ltd., OXford (1987)
Shelukhin V.V.: A shear problem for the compressible Navier–Stokes equations. Int. J. Non-Linear Mech. 33, 247–257 (1998)
Shelukhin V.V.: The limit of zero shear viscosity for compressible fluids. Arch. Ration. Mech. Anal. 143, 357–374 (1998)
Shelukhin V.V.: Vanishing shear viscosity in a free boundary problem for the equations of compressible fluids. J. Differ. Equ. 167, 73–86 (2000)
Tan Z., Yang T., Zhao H.J., Zou Q.Y.: Global solutions to the one-dimensional compressible Navier–Stokes–Poisson equations with large data. SIAM J. Math. Anal. 45, 547–571 (2013)
Yao L., Zhang T., Zhu C.J.: Boundary layers for compressible Navier–Stokes equations with density dependent viscosity and cylindrical symmetry. Ann. I. H. Poincaré 28, 677–709 (2011)
Zek’dovich, Y.B., Raizer, Y. P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, vol. II. Academic Press, New York, (1967)
Zhang J.W., Jiang S., Xie F.: Global weak solutions of an initial boundary value problem for screw pinches in plasma physics. Math. Models Methods Appl. Sci. 19, 833–875 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Lin
Rights and permissions
About this article
Cite this article
Qin, X., Yang, T., Yao, Za. et al. Vanishing Shear Viscosity and Boundary Layer for the Navier–Stokes Equations with Cylindrical Symmetry. Arch Rational Mech Anal 216, 1049–1086 (2015). https://doi.org/10.1007/s00205-014-0826-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-014-0826-x