# American Institute of Mathematical Sciences

October  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 and 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), 110-146. 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), 1-87. 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), 1143-1174. [4] S. Kawashima, "Systems of a Hyperbilica Parabolic Type with Applications to the Equations of Magneto Hydrodynamics," Ph.D thesis, Kyoto University, 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), 169-194. doi: 10.1017/S0308210500018308. [6] A. Majda and R. L. Pego, Stable viscosity matrices for systems of conservation laws, J. Differential Equations, 56 (1985), 229-262. 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), 529-575. 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), 445-464. doi: 10.1007/BF01214738. [9] J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187. 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), 2991-3008. doi: 10.1090/S0002-9947-03-03279-3. [11] F. Rousset, Characteristic boundary layers in real vanishing viscosity limits, J. Differential Equations, 210 (2005), 25-64. 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, Cambridge University Press, Cambridge, 2000. [13] D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit, Comm. Math. Phys., 221 (2001), 267-292. 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), 193-253. doi: 10.2977/prims/1195190106. [15] Z. Xin, Viscous boundary layers and their stability. I, J. Partial Differential Equations, 11 (1998), 97-124. [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), 479-541. 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 "First International Congress of Chinese Mathematicians" (Beijing, 1998), AMS/IP Stud. Adv. Math., 20, Amer. Math. Soc., Providence, RI, (2001), 159-170.

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), 110-146. 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), 1-87. 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), 1143-1174. [4] S. Kawashima, "Systems of a Hyperbilica Parabolic Type with Applications to the Equations of Magneto Hydrodynamics," Ph.D thesis, Kyoto University, 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), 169-194. doi: 10.1017/S0308210500018308. [6] A. Majda and R. L. Pego, Stable viscosity matrices for systems of conservation laws, J. Differential Equations, 56 (1985), 229-262. 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), 529-575. 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), 445-464. doi: 10.1007/BF01214738. [9] J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187. 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), 2991-3008. doi: 10.1090/S0002-9947-03-03279-3. [11] F. Rousset, Characteristic boundary layers in real vanishing viscosity limits, J. Differential Equations, 210 (2005), 25-64. 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, Cambridge University Press, Cambridge, 2000. [13] D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit, Comm. Math. Phys., 221 (2001), 267-292. 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), 193-253. doi: 10.2977/prims/1195190106. [15] Z. Xin, Viscous boundary layers and their stability. I, J. Partial Differential Equations, 11 (1998), 97-124. [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), 479-541. 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 "First International Congress of Chinese Mathematicians" (Beijing, 1998), AMS/IP Stud. Adv. Math., 20, Amer. Math. Soc., Providence, RI, (2001), 159-170.
 [1] Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041 [2] Ping Chen, Ting Zhang. A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients. Communications on Pure and Applied Analysis, 2008, 7 (4) : 987-1016. doi: 10.3934/cpaa.2008.7.987 [3] Yuming Qin, Lan Huang, Shuxian Deng, Zhiyong Ma, Xiaoke Su, Xinguang Yang. Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 163-192. doi: 10.3934/dcdss.2009.2.163 [4] Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Networks and Heterogeneous Media, 2012, 7 (4) : 741-766. doi: 10.3934/nhm.2012.7.741 [5] Wenjun Wang, Lei Yao. Spherically symmetric Navier-Stokes equations with degenerate viscosity coefficients and vacuum. Communications on Pure and Applied Analysis, 2010, 9 (2) : 459-481. doi: 10.3934/cpaa.2010.9.459 [6] Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153 [7] Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353 [8] Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673 [9] Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 [10] Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic and Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409 [11] Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201 [12] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [13] Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2811-2838. doi: 10.3934/cpaa.2021080 [14] Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277 [15] Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure and Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373 [16] Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228 [17] Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113 [18] Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769 [19] Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355 [20] Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic and Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75

2021 Impact Factor: 1.497