May  2016, 36(5): 2673-2709. doi: 10.3934/dcds.2016.36.2673

The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions

1. 

Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005 Paris, France

Received  November 2014 Revised  September 2015 Published  October 2015

We obtain existence and conormal Sobolev regularity of strong solutions to the 3D compressible isentropic Navier-Stokes system on the half-space with a Navier boundary condition, over a time that is uniform with respect to the viscosity parameters when these are small. These solutions then converge globally in space and strongly in $L^2$ towards the solution of the compressible isentropic Euler system when the viscosity parameters go to zero.
Citation: Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673
References:
[1]

F. Ancona and S. Bianchini, Vanishing viscosity solutions of hyperbolic systems of conservation laws with boundary,, in , (2005), 13. doi: 10.1142/9789812773616_0003.

[2]

C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux,, J. Math. Anal. Appl., 40 (1972), 769. doi: 10.1016/0022-247X(72)90019-4.

[3]

C. Bardos, F. Golse and L. Paillard, The incompressible Euler limit of the Boltzmann equation with accommodation boundary condition,, Commun. Math. Sci., 10 (2012), 159. doi: 10.4310/CMS.2012.v10.n1.a9.

[4]

H. Beirão da Veiga and F. Crispo, The 3-D inviscid limit result under slip boundary conditions. A negative answer,, J. Math. Fluid Mech., 14 (2012), 55. doi: 10.1007/s00021-010-0047-5.

[5]

D. Bresch, B. Desjardins and D. Gérard-Varet, On compressible Navier-Stokes equations with density dependent viscosities in bounded domains,, J. Math. Pures Appl. (9), 87 (2007), 227. doi: 10.1016/j.matpur.2006.10.010.

[6]

D. Bucur, A.-L. Dalibard and D. Gérard-Varet, Wall laws for viscous fluids near rough surfaces,, in Mathematical and Numerical Approaches for Multiscale Problem, (2012), 117. doi: 10.1051/proc/201237003.

[7]

M. Bulíček, J. Málek and K. R. Rajagopal, Navier's slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity,, Indiana Univ. Math. J., 56 (2007), 51. doi: 10.1512/iumj.2007.56.2997.

[8]

T. Clopeau, A. Mikelić and R. Robert, On the vanishing viscosity limit for the $2D$ incompressible Navier-Stokes equations with the friction type boundary conditions,, Nonlinearity, 11 (1998), 1625. doi: 10.1088/0951-7715/11/6/011.

[9]

R. Danchin, A survey on Fourier analysis methods for solving the compressible Navier-Stokes equations,, Sci. China Math., 55 (2012), 245. doi: 10.1007/s11425-011-4357-8.

[10]

B. Desjardins and C.-K. Lin, A survey of the compressible Navier-Stokes equations,, Taiwanese J. Math., 3 (1999), 123.

[11]

E. Feireisl and A. Novotný, Inviscid incompressible limits of the full Navier-Stokes-Fourier system,, Comm. Math. Phys., 321 (2013), 605. doi: 10.1007/s00220-013-1691-4.

[12]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, J. Math. Fluid Mech., 3 (2001), 358. doi: 10.1007/PL00000976.

[13]

D. Gérard-Varet and N. Masmoudi, Relevance of the slip condition for fluid flows near an irregular boundary,, Comm. Math. Phys., 295 (2010), 99. doi: 10.1007/s00220-009-0976-0.

[14]

O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique,, Comm. Partial Differential Equations, 15 (1990), 595. doi: 10.1080/03605309908820701.

[15]

O. Guès, G. Métivier, M. Williams and K. Zumbrun, Existence and stability 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.

[16]

D. Hoff, Compressible flow in a half-space with Navier boundary conditions,, J. Math. Fluid Mech., 7 (2005), 315. doi: 10.1007/s00021-004-0123-9.

[17]

L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems,, Ann. of Math. (2), 83 (1966), 129. doi: 10.2307/1970473.

[18]

F. Huang, Y. Wang and T. Yang, Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to a Riemann problem,, Arch. Ration. Mech. Anal., 203 (2012), 379. doi: 10.1007/s00205-011-0450-y.

[19]

D. Iftimie and G. Planas, Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions,, Nonlinearity, 19 (2006), 899. doi: 10.1088/0951-7715/19/4/007.

[20]

D. Iftimie and F. Sueur, Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions,, Arch. Ration. Mech. Anal., 199 (2011), 145. doi: 10.1007/s00205-010-0320-z.

[21]

W. Jäger and A. Mikelić, On the roughness-induced effective boundary conditions for an incompressible viscous flow,, J. Differential Equations, 170 (2001), 96. doi: 10.1006/jdeq.2000.3814.

[22]

J. P. Kelliher, Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane,, SIAM J. Math. Anal., 38 (2006), 210. doi: 10.1137/040612336.

[23]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid,, J. Math. Pures Appl. (9), 77 (1998), 585. doi: 10.1016/S0021-7824(98)80139-6.

[24]

P.-L. Lions, Existence globale de solutions pour les équations de Navier-Stokes compressibles isentropiques,, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1335.

[25]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models,, Oxford Lecture Series in Mathematics and its Applications, (1998).

[26]

N. Masmoudi and F. Rousset, Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations,, preprint, ().

[27]

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.

[28]

N. Masmoudi and L. Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in a bounded domain,, Comm. Pure Appl. Math., 56 (2003), 1263. doi: 10.1002/cpa.10095.

[29]

G. Métivier and K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems,, Mem. Amer. Math. Soc., 175 (2005). doi: 10.1090/memo/0826.

[30]

J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France, 90 (1962), 487.

[31]

C.-L.-M.-H. Navier, Mémoire sur les lois du mouvement des fluides,, Mém. Acad. Roy. Sci. Inst. France, 6 (1823), 389.

[32]

M. Paddick, Stability and instability of Navier boundary layers,, Differential Integral Equations, 27 (2014), 893.

[33]

D. Pal, N. Rudraiah and R. Devanathan, The effects of slip velocity at a membrane surface on blood flow in the microcirculation,, J. Math. Biol., 26 (1988), 705. doi: 10.1007/BF00276149.

[34]

T. Qian, X.-P. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows,, Phys. Rev. E, 68 (2003). doi: 10.1103/PhysRevE.68.016306.

[35]

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.

[36]

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

[37]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit,, Comm. Math. Phys., 104 (1986), 49. doi: 10.1007/BF01210792.

[38]

P. Secchi, Well-posedness of characteristic symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 134 (1996), 155. doi: 10.1007/BF00379552.

[39]

P. Secchi, Some properties of anisotropic Sobolev spaces,, Arch. Math. (Basel), 75 (2000), 207. doi: 10.1007/s000130050494.

[40]

V. A. Solonnikov, The solvability of the initial-boundary value problem for the equations of motion of a viscous compressible fluid. Investigations on linear operators and theory of functions, VI,, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 56 (1976), 128.

[41]

F. Sueur, On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain,, J. Math. Fluid Mech., 16 (2014), 163. doi: 10.1007/s00021-013-0145-2.

[42]

X.-P. Wang, Y.-G. Wang and Z. Xin, Boundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limit,, Commun. Math. Sci., 8 (2010), 965. doi: 10.4310/CMS.2010.v8.n4.a10.

[43]

Y.-G. Wang and M. Williams, The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions,, Ann. Inst. Fourier (Grenoble), 62 (2012), 2257. doi: 10.5802/aif.2749.

[44]

Y. Xiao and Z. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition,, Comm. Pure Appl. Math., 60 (2007), 1027. doi: 10.1002/cpa.20187.

[45]

Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,, Comm. Pure Appl. Math., 51 (1998), 229. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[46]

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>3.0.CO;2-1.

show all references

References:
[1]

F. Ancona and S. Bianchini, Vanishing viscosity solutions of hyperbolic systems of conservation laws with boundary,, in , (2005), 13. doi: 10.1142/9789812773616_0003.

[2]

C. Bardos, Existence et unicité de la solution de l'équation d'Euler en dimension deux,, J. Math. Anal. Appl., 40 (1972), 769. doi: 10.1016/0022-247X(72)90019-4.

[3]

C. Bardos, F. Golse and L. Paillard, The incompressible Euler limit of the Boltzmann equation with accommodation boundary condition,, Commun. Math. Sci., 10 (2012), 159. doi: 10.4310/CMS.2012.v10.n1.a9.

[4]

H. Beirão da Veiga and F. Crispo, The 3-D inviscid limit result under slip boundary conditions. A negative answer,, J. Math. Fluid Mech., 14 (2012), 55. doi: 10.1007/s00021-010-0047-5.

[5]

D. Bresch, B. Desjardins and D. Gérard-Varet, On compressible Navier-Stokes equations with density dependent viscosities in bounded domains,, J. Math. Pures Appl. (9), 87 (2007), 227. doi: 10.1016/j.matpur.2006.10.010.

[6]

D. Bucur, A.-L. Dalibard and D. Gérard-Varet, Wall laws for viscous fluids near rough surfaces,, in Mathematical and Numerical Approaches for Multiscale Problem, (2012), 117. doi: 10.1051/proc/201237003.

[7]

M. Bulíček, J. Málek and K. R. Rajagopal, Navier's slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity,, Indiana Univ. Math. J., 56 (2007), 51. doi: 10.1512/iumj.2007.56.2997.

[8]

T. Clopeau, A. Mikelić and R. Robert, On the vanishing viscosity limit for the $2D$ incompressible Navier-Stokes equations with the friction type boundary conditions,, Nonlinearity, 11 (1998), 1625. doi: 10.1088/0951-7715/11/6/011.

[9]

R. Danchin, A survey on Fourier analysis methods for solving the compressible Navier-Stokes equations,, Sci. China Math., 55 (2012), 245. doi: 10.1007/s11425-011-4357-8.

[10]

B. Desjardins and C.-K. Lin, A survey of the compressible Navier-Stokes equations,, Taiwanese J. Math., 3 (1999), 123.

[11]

E. Feireisl and A. Novotný, Inviscid incompressible limits of the full Navier-Stokes-Fourier system,, Comm. Math. Phys., 321 (2013), 605. doi: 10.1007/s00220-013-1691-4.

[12]

E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations,, J. Math. Fluid Mech., 3 (2001), 358. doi: 10.1007/PL00000976.

[13]

D. Gérard-Varet and N. Masmoudi, Relevance of the slip condition for fluid flows near an irregular boundary,, Comm. Math. Phys., 295 (2010), 99. doi: 10.1007/s00220-009-0976-0.

[14]

O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique,, Comm. Partial Differential Equations, 15 (1990), 595. doi: 10.1080/03605309908820701.

[15]

O. Guès, G. Métivier, M. Williams and K. Zumbrun, Existence and stability 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.

[16]

D. Hoff, Compressible flow in a half-space with Navier boundary conditions,, J. Math. Fluid Mech., 7 (2005), 315. doi: 10.1007/s00021-004-0123-9.

[17]

L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems,, Ann. of Math. (2), 83 (1966), 129. doi: 10.2307/1970473.

[18]

F. Huang, Y. Wang and T. Yang, Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to a Riemann problem,, Arch. Ration. Mech. Anal., 203 (2012), 379. doi: 10.1007/s00205-011-0450-y.

[19]

D. Iftimie and G. Planas, Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions,, Nonlinearity, 19 (2006), 899. doi: 10.1088/0951-7715/19/4/007.

[20]

D. Iftimie and F. Sueur, Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions,, Arch. Ration. Mech. Anal., 199 (2011), 145. doi: 10.1007/s00205-010-0320-z.

[21]

W. Jäger and A. Mikelić, On the roughness-induced effective boundary conditions for an incompressible viscous flow,, J. Differential Equations, 170 (2001), 96. doi: 10.1006/jdeq.2000.3814.

[22]

J. P. Kelliher, Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane,, SIAM J. Math. Anal., 38 (2006), 210. doi: 10.1137/040612336.

[23]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid,, J. Math. Pures Appl. (9), 77 (1998), 585. doi: 10.1016/S0021-7824(98)80139-6.

[24]

P.-L. Lions, Existence globale de solutions pour les équations de Navier-Stokes compressibles isentropiques,, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1335.

[25]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models,, Oxford Lecture Series in Mathematics and its Applications, (1998).

[26]

N. Masmoudi and F. Rousset, Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations,, preprint, ().

[27]

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.

[28]

N. Masmoudi and L. Saint-Raymond, From the Boltzmann equation to the Stokes-Fourier system in a bounded domain,, Comm. Pure Appl. Math., 56 (2003), 1263. doi: 10.1002/cpa.10095.

[29]

G. Métivier and K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems,, Mem. Amer. Math. Soc., 175 (2005). doi: 10.1090/memo/0826.

[30]

J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France, 90 (1962), 487.

[31]

C.-L.-M.-H. Navier, Mémoire sur les lois du mouvement des fluides,, Mém. Acad. Roy. Sci. Inst. France, 6 (1823), 389.

[32]

M. Paddick, Stability and instability of Navier boundary layers,, Differential Integral Equations, 27 (2014), 893.

[33]

D. Pal, N. Rudraiah and R. Devanathan, The effects of slip velocity at a membrane surface on blood flow in the microcirculation,, J. Math. Biol., 26 (1988), 705. doi: 10.1007/BF00276149.

[34]

T. Qian, X.-P. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows,, Phys. Rev. E, 68 (2003). doi: 10.1103/PhysRevE.68.016306.

[35]

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.

[36]

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

[37]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit,, Comm. Math. Phys., 104 (1986), 49. doi: 10.1007/BF01210792.

[38]

P. Secchi, Well-posedness of characteristic symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 134 (1996), 155. doi: 10.1007/BF00379552.

[39]

P. Secchi, Some properties of anisotropic Sobolev spaces,, Arch. Math. (Basel), 75 (2000), 207. doi: 10.1007/s000130050494.

[40]

V. A. Solonnikov, The solvability of the initial-boundary value problem for the equations of motion of a viscous compressible fluid. Investigations on linear operators and theory of functions, VI,, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 56 (1976), 128.

[41]

F. Sueur, On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain,, J. Math. Fluid Mech., 16 (2014), 163. doi: 10.1007/s00021-013-0145-2.

[42]

X.-P. Wang, Y.-G. Wang and Z. Xin, Boundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limit,, Commun. Math. Sci., 8 (2010), 965. doi: 10.4310/CMS.2010.v8.n4.a10.

[43]

Y.-G. Wang and M. Williams, The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions,, Ann. Inst. Fourier (Grenoble), 62 (2012), 2257. doi: 10.5802/aif.2749.

[44]

Y. Xiao and Z. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition,, Comm. Pure Appl. Math., 60 (2007), 1027. doi: 10.1002/cpa.20187.

[45]

Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,, Comm. Pure Appl. Math., 51 (1998), 229. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[46]

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>3.0.CO;2-1.

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