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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 |
References:
[1] |
F. Ancona and S. Bianchini, Vanishing viscosity solutions of hyperbolic systems of conservation laws with boundary, in "WASCOM 2005''-13th Conference on Waves and Stability in Continuous Media, World Sci. Publ., Hackensack, NJ, 2006, 13-21.
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), 769-790.
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), 159-190.
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), 55-59.
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), 227-235.
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, ESAIM Proc., 37, EDP Sci., Les Ulis, 2012, 117-135.
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), 51-85.
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), 1625-1636.
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), 245-275.
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), 123-137. |
[11] |
E. Feireisl and A. Novotný, Inviscid incompressible limits of the full Navier-Stokes-Fourier system, Comm. Math. Phys., 321 (2013), 605-628.
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), 358-392.
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), 99-137.
doi: 10.1007/s00220-009-0976-0. |
[14] |
O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations, 15 (1990), 595-645.
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), 1-87.
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), 315-338.
doi: 10.1007/s00021-004-0123-9. |
[17] |
L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math. (2), 83 (1966), 129-209.
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), 379-413.
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), 899-918.
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), 145-175.
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), 96-122.
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), 210-232 (electronic).
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), 585-627.
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), 1335-1340. |
[25] |
P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 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), 529-575.
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), 1263-1293.
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), vi+107.
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), 487-497. |
[31] |
C.-L.-M.-H. Navier, Mémoire sur les lois du mouvement des fluides, Mém. Acad. Roy. Sci. Inst. France, 6 (1823), 389-410. Available from: http://gallica.bnf.fr/ark:/12148/bpt6k3221x/f577. |
[32] |
M. Paddick, Stability and instability of Navier boundary layers, Differential Integral Equations, 27 (2014), 893-930. Available from: http://projecteuclid.org/euclid.die/1404230050. |
[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), 705-712.
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), 016306.
doi: 10.1103/PhysRevE.68.016306. |
[35] |
J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187.
doi: 10.1090/S0002-9947-1985-0797053-4. |
[36] |
F. Rousset, Characteristic boundary layers in real vanishing viscosity limits, J. Differential Equations, 210 (2005), 25-64.
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), 49-75. Available from: http://projecteuclid.org/getRecord?id=euclid.cmp/1104114932.
doi: 10.1007/BF01210792. |
[38] |
P. Secchi, Well-posedness of characteristic symmetric hyperbolic systems, Arch. Rational Mech. Anal., 134 (1996), 155-197.
doi: 10.1007/BF00379552. |
[39] |
P. Secchi, Some properties of anisotropic Sobolev spaces, Arch. Math. (Basel), 75 (2000), 207-216.
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), 128-142, 197. |
[41] |
F. Sueur, On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain, J. Math. Fluid Mech., 16 (2014), 163-178.
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), 965-998. Available from: http://projecteuclid.org/getRecord?id=euclid.cms/1288725268.
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), 2257-2314 (2013).
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), 1027-1055.
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), 229-240.
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), 479-541.
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 "WASCOM 2005''-13th Conference on Waves and Stability in Continuous Media, World Sci. Publ., Hackensack, NJ, 2006, 13-21.
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), 769-790.
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), 159-190.
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), 55-59.
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), 227-235.
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, ESAIM Proc., 37, EDP Sci., Les Ulis, 2012, 117-135.
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), 51-85.
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), 1625-1636.
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), 245-275.
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), 123-137. |
[11] |
E. Feireisl and A. Novotný, Inviscid incompressible limits of the full Navier-Stokes-Fourier system, Comm. Math. Phys., 321 (2013), 605-628.
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), 358-392.
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), 99-137.
doi: 10.1007/s00220-009-0976-0. |
[14] |
O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique, Comm. Partial Differential Equations, 15 (1990), 595-645.
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), 1-87.
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), 315-338.
doi: 10.1007/s00021-004-0123-9. |
[17] |
L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math. (2), 83 (1966), 129-209.
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), 379-413.
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), 899-918.
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), 145-175.
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), 96-122.
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), 210-232 (electronic).
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), 585-627.
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), 1335-1340. |
[25] |
P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 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), 529-575.
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), 1263-1293.
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), vi+107.
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), 487-497. |
[31] |
C.-L.-M.-H. Navier, Mémoire sur les lois du mouvement des fluides, Mém. Acad. Roy. Sci. Inst. France, 6 (1823), 389-410. Available from: http://gallica.bnf.fr/ark:/12148/bpt6k3221x/f577. |
[32] |
M. Paddick, Stability and instability of Navier boundary layers, Differential Integral Equations, 27 (2014), 893-930. Available from: http://projecteuclid.org/euclid.die/1404230050. |
[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), 705-712.
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), 016306.
doi: 10.1103/PhysRevE.68.016306. |
[35] |
J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187.
doi: 10.1090/S0002-9947-1985-0797053-4. |
[36] |
F. Rousset, Characteristic boundary layers in real vanishing viscosity limits, J. Differential Equations, 210 (2005), 25-64.
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), 49-75. Available from: http://projecteuclid.org/getRecord?id=euclid.cmp/1104114932.
doi: 10.1007/BF01210792. |
[38] |
P. Secchi, Well-posedness of characteristic symmetric hyperbolic systems, Arch. Rational Mech. Anal., 134 (1996), 155-197.
doi: 10.1007/BF00379552. |
[39] |
P. Secchi, Some properties of anisotropic Sobolev spaces, Arch. Math. (Basel), 75 (2000), 207-216.
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), 128-142, 197. |
[41] |
F. Sueur, On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain, J. Math. Fluid Mech., 16 (2014), 163-178.
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), 965-998. Available from: http://projecteuclid.org/getRecord?id=euclid.cms/1288725268.
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), 2257-2314 (2013).
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), 1027-1055.
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), 229-240.
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), 479-541.
doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.0.CO;2-1. |
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