Boundary layers for compressible Navier–Stokes equations with density-dependent viscosity and cylindrical symmetry

Yao, Lei; Zhang, Ting; Zhu, Changjiang

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 5, pp. 677-709.

In this paper, we consider the zero shear viscosity limit for the Navier–Stokes equations of compressible flows with density-dependent viscosity coefficient and cylindrical symmetry. The boundary layer effect as the shear viscosity $\mu =ϵ{\rho }^{\theta }$ goes to zero (in fact, $ϵ\to 0$ in this paper, which implies $\mu \to 0$) is studied. We prove that the boundary layer thickness is of the order $O\left({ϵ}^{\alpha }\right)$, where $0<\alpha <\frac{1}{2}$ for the constant initial data and $0<\alpha <\frac{1}{4}$ for the general initial data, which extend the result in Frid and Shelukhin (1999) [4] to the case of density-dependent viscosity coefficient.

Classification : 76N20, 35B40, 35Q30, 76N10, 76N17
Mots clés : Navier–Stokes equations, Density-dependent viscosity, Cylindrical symmetry, Zero shear viscosity limit, Boundary layers, BL-thickness

[1] S. Chapman, T.G. Cowling, The Mathematical Theory of Non-uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, London (1970) | MR | Zbl

[2] J. Fan, S. Jiang, Zero shear viscosity limit for the Navier–Stokes equations of compressible isentropic fluids with cylindric symmetry, Rend. Semin. Mat. Univ. Politec. Torino 65 (2007), 35-52 | MR | Zbl

[3] P.C. Fife, Considerations regarding the mathematical basis for Prandtlʼs boundary layer theory, Arch. Ration. Mech. Anal. 28 (1968), 184-216 | MR | Zbl

[4] H. Frid, V. Shelukhin, Boundary layers for the Navier–Stokes equations of compressible fluids, Comm. Math. Phys. 208 (1999), 309-330 | MR | Zbl

[5] H. Frid, V. Shelukhin, Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry, SIAM J. Math. Anal. 31 (2000), 1144-1156 | MR | Zbl

[6] H. Grad, Asymptotic theory of the Boltzmann equation, II, Rarefied Gas Dynamics, vol. 1, Academic Press, New York (1963), 26-59 | MR | Zbl

[7] S. Jiang, J. Zhang, Boundary layers for the Navier–Stokes equations of compressible heat-conducting flows with cylindrical symmetry, SIAM J. Math. Anal. 41 (2009), 237-268 | MR | Zbl

[8] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford (1987) | MR | Zbl

[9] O.A. Oleinik, V.N. Samokhin, Mathematical Models in Boundary Layer, Chapman & Hall/CRC, London (1999) | Zbl

[10] L. Rosenhead, A discussion on the first and second viscosities of fluids. Introduction. The second coefficient of viscosity: a brief review of fundamentals, Proc. R. Soc. Lond. Ser. A 226 (1954), 1-6 | MR

[11] F. Rousset, Stability of small amplitude boundary layers for mixed hyperbolic–parabolic systems, Trans. Amer. Math. Soc. 355 (2003), 2991-3008 | MR | Zbl

[12] M. Sammartino, R.E. Caflisch, Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. I: Existence for Euler and Prandtl equations, Comm. Math. Phys. 192 (1998), 433-461 | MR | Zbl

[13] M. Sammartino, R.E. Caflisch, Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. II: Construction of the Navier–Stokes solution, Comm. Math. Phys. 192 (1998), 463-491 | MR | Zbl

[14] H. Schlichting, Boundary Layer Theory, McGraw–Hill Company, London, New York (1979) | MR | Zbl

[15] P. Secchi, Existence theorems for compressible viscous fluids having zero shear viscosity, Rend. Semin. Mat. Univ. Padova 71 (1984), 73-102 | EuDML | Numdam | MR | Zbl

[16] D. Serre, K. Zumbrun, Boundary layer stability in real vanishing viscosity limit, Comm. Math. Phys. 221 (2001), 267-292 | MR | Zbl

[17] J. Serrin, Mathematical Principles of Classical Fluid Mechanics, Handbuch der Physik vol. 8/1, Springer (1959) | MR

[18] J. Serrin, On the mathematical basis for Prandtlʼs boundary layer theory: An example, Arch. Ration. Mech. Anal. 28 (1968), 217-225 | MR | Zbl

[19] V.V. Shelukhin, A shear flow problem for the compressible Navier–Stokes equations, Int.. J. Non-Linear Mech. 33 (1998), 247-257 | MR | Zbl

[20] V. Shelukhin, The limit of zero shear viscosity for compressible fluids, Arch. Ration. Mech. Anal. 143 (1998), 357-374 | MR | Zbl

[21] V. Shelukhin, Vanishing shear viscosity in a free-boundary problem for the equations of compressible fluids, J. Differential Equations 167 (2000), 73-86 | MR | Zbl

[22] I. Straškraba, Global analysis of 1-D Navier–Stokes equations with density dependent viscosity, H. Amann, et al. (ed.), Navier–Stokes Equations and Related Nonlinear Problems, VSP, Utrecht (1998), 371-389 | MR | Zbl

[23] I. Straškraba, A. Zlotnik, Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity, Z. Angew. Math. Phys. 54 (2003), 593-607 | MR | Zbl

[24] R. Temam, X.M. Wang, Asymptotic analysis of the linearized Navier–Stokes equations in a channel, Differential Integral Equations 8 (1995), 1591-1618 | MR | Zbl

[25] G. Tian, Z.P. Xin, Gradient estimation on Navier–Stokes equations, Comm. Anal. Geom. 7 (1999), 221-257 | MR | Zbl

[26] Y.G. Wang, Z.P. Xin, Zero-viscosity limit of the linearized compressible Navier–Stokes equations with highly oscillatory forces in the half-plane, SIAM J. Math. Anal. 37 (2005), 1256-1298 | MR | Zbl

[27] Z.P. Xin, Viscous boundary layers and their stability, I, J. Partial Differential Equations 11 (1998), 97-124 | MR | Zbl

[28] Z.P. Xin, T. Yanagisawa, Zero-viscosity limit of the linearized Navier–Stokes equations for a compressible viscous fluid in the half-plane, Comm. Pure Appl. Math. 52 (1999), 479-541 | MR | Zbl

[29] A. Zlotnik, Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations, Differ. Equ. 36 (2000), 701-716 | MR | Zbl