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

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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

Matthew Paddick ^1,

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

Abstract
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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.

Keywords: conormal Sobolev spaces, Compressible Navier-Stokes equations, inviscid limit problem., Navier (slip) boundary condition.

Mathematics Subject Classification: Primary: 35Q30, 76N10; Secondary: 76N2.

Citation: 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

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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[19]	Jean-Pierre Raymond. Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1537-1564. doi: 10.3934/dcdsb.2010.14.1537
[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

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