2012, 17(7): 2595-2613. doi: 10.3934/dcdsb.2012.17.2595

Stability of boundary layers for the inflow compressible Navier-Stokes equations

1. 

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

2. 

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received  April 2011 Revised  January 2012 Published  July 2012

In this paper, we consider the boundary layer stability of the one-dimensional isentropic compressible Navier-Stokes equations with an inflow boundary condition. We assume only one of the two characteristics to the corresponding Euler equations is negative up to some small time. We prove the existence of the boundary layers, then instead of using the skew symmetric matrix, we give a higher convergence rate of the approximate solution than the previous results by a standard energy method as long as the strength of the boundary layers is suitably small.
Citation: Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595
References:
[1]

E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems,, J. Differential Equations, 143 (1998), 110. doi: 10.1006/jdeq.1997.3364.

[2]

O. Guès, G. Métivier, M. Williams and K. Zumbrun, Existence andstability of noncharacteristic boundary layers for the compressible Navier-Stokes and viscous MHD equations,, Arch. Ration. Mech. Anal., 197 (2010), 1. doi: 10.1007/s00205-009-0277-y.

[3]

J. Wang, Boundary layers for compressible Navier-Stokes equations with outflow boundary condition,, J. Differential Equations, 248 (2010), 1143.

[4]

S. Kawashima, "Systems of a Hyperbilica Parabolic Type with Applications to the Equations of Magneto Hydrodynamics,", Ph.D thesis, (1983).

[5]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications,, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169. doi: 10.1017/S0308210500018308.

[6]

A. Majda and R. L. Pego, Stable viscosity matrices for systems of conservation laws,, J. Differential Equations, 56 (1985), 229. doi: 10.1016/0022-0396(85)90107-X.

[7]

N. Masmoudi and F. Rousset, Uniform regularity for the Navier-Stokes equation with Navier boundary condition,, Arch. Ration. Mech. Anal., 203 (2012), 529. doi: 10.1007/s00205-011-0456-5.

[8]

A. Matsumura and T. Nishida, Initial-boundary value problems forthe equations of motion of compressible viscous and heat-conductive fluids,, Comm. Math. Phys., 89 (1983), 445. doi: 10.1007/BF01214738.

[9]

J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity,, Trans. Amer. Math. Soc., 291 (1985), 167. doi: 10.1090/S0002-9947-1985-0797053-4.

[10]

F. Rousset, Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems,, Trans. Amer. Math. Soc., 355 (2003), 2991. doi: 10.1090/S0002-9947-03-03279-3.

[11]

F. Rousset, Characteristic boundary layers in real vanishing viscosity limits,, J. Differential Equations, 210 (2005), 25. doi: 10.1016/j.jde.2004.10.004.

[12]

D. Serre, "Systems of Conservation Laws. 2. Geometric Structures, Oscillations, and Initial-Boundary Value Problems,", Translated from the 1996 French original by I. N. Sneddon, (1996).

[13]

D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit,, Comm. Math. Phys., 221 (2001), 267. doi: 10.1007/s002200100486.

[14]

A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion,, Publ. RIMS. Kyoto Univ., 13 (1977), 193. doi: 10.2977/prims/1195190106.

[15]

Z. Xin, Viscous boundary layers and their stability. I,, J. Partial Differential Equations, 11 (1998), 97.

[16]

Z. Xin and 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. doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.3.CO;2-T.

[17]

Z. Xin, On the behavior of solutions to the compressible Navier-Stokes equations,, in, 20 (2001), 159.

show all references

References:
[1]

E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems,, J. Differential Equations, 143 (1998), 110. doi: 10.1006/jdeq.1997.3364.

[2]

O. Guès, G. Métivier, M. Williams and K. Zumbrun, Existence andstability of noncharacteristic boundary layers for the compressible Navier-Stokes and viscous MHD equations,, Arch. Ration. Mech. Anal., 197 (2010), 1. doi: 10.1007/s00205-009-0277-y.

[3]

J. Wang, Boundary layers for compressible Navier-Stokes equations with outflow boundary condition,, J. Differential Equations, 248 (2010), 1143.

[4]

S. Kawashima, "Systems of a Hyperbilica Parabolic Type with Applications to the Equations of Magneto Hydrodynamics,", Ph.D thesis, (1983).

[5]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications,, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169. doi: 10.1017/S0308210500018308.

[6]

A. Majda and R. L. Pego, Stable viscosity matrices for systems of conservation laws,, J. Differential Equations, 56 (1985), 229. doi: 10.1016/0022-0396(85)90107-X.

[7]

N. Masmoudi and F. Rousset, Uniform regularity for the Navier-Stokes equation with Navier boundary condition,, Arch. Ration. Mech. Anal., 203 (2012), 529. doi: 10.1007/s00205-011-0456-5.

[8]

A. Matsumura and T. Nishida, Initial-boundary value problems forthe equations of motion of compressible viscous and heat-conductive fluids,, Comm. Math. Phys., 89 (1983), 445. doi: 10.1007/BF01214738.

[9]

J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity,, Trans. Amer. Math. Soc., 291 (1985), 167. doi: 10.1090/S0002-9947-1985-0797053-4.

[10]

F. Rousset, Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems,, Trans. Amer. Math. Soc., 355 (2003), 2991. doi: 10.1090/S0002-9947-03-03279-3.

[11]

F. Rousset, Characteristic boundary layers in real vanishing viscosity limits,, J. Differential Equations, 210 (2005), 25. doi: 10.1016/j.jde.2004.10.004.

[12]

D. Serre, "Systems of Conservation Laws. 2. Geometric Structures, Oscillations, and Initial-Boundary Value Problems,", Translated from the 1996 French original by I. N. Sneddon, (1996).

[13]

D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit,, Comm. Math. Phys., 221 (2001), 267. doi: 10.1007/s002200100486.

[14]

A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion,, Publ. RIMS. Kyoto Univ., 13 (1977), 193. doi: 10.2977/prims/1195190106.

[15]

Z. Xin, Viscous boundary layers and their stability. I,, J. Partial Differential Equations, 11 (1998), 97.

[16]

Z. Xin and 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. doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.3.CO;2-T.

[17]

Z. Xin, On the behavior of solutions to the compressible Navier-Stokes equations,, in, 20 (2001), 159.

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