# American Institute of Mathematical Sciences

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] B. Desjardins and C.-K. Lin, A survey of the compressible Navier-Stokes equations,, Taiwanese J. Math., 3 (1999), 123. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique,, Comm. Partial Differential Equations, 15 (1990), 595. doi: 10.1080/03605309908820701. Google Scholar [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. Google Scholar [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. Google Scholar [17] L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems,, Ann. of Math. (2), 83 (1966), 129. doi: 10.2307/1970473. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [25] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models,, Oxford Lecture Series in Mathematics and its Applications, (1998). Google Scholar [26] N. Masmoudi and F. Rousset, Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations,, preprint, (). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France, 90 (1962), 487. Google Scholar [31] C.-L.-M.-H. Navier, Mémoire sur les lois du mouvement des fluides,, Mém. Acad. Roy. Sci. Inst. France, 6 (1823), 389. Google Scholar [32] M. Paddick, Stability and instability of Navier boundary layers,, Differential Integral Equations, 27 (2014), 893. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [38] P. Secchi, Well-posedness of characteristic symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 134 (1996), 155. doi: 10.1007/BF00379552. Google Scholar [39] P. Secchi, Some properties of anisotropic Sobolev spaces,, Arch. Math. (Basel), 75 (2000), 207. doi: 10.1007/s000130050494. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] B. Desjardins and C.-K. Lin, A survey of the compressible Navier-Stokes equations,, Taiwanese J. Math., 3 (1999), 123. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique,, Comm. Partial Differential Equations, 15 (1990), 595. doi: 10.1080/03605309908820701. Google Scholar [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. Google Scholar [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. Google Scholar [17] L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems,, Ann. of Math. (2), 83 (1966), 129. doi: 10.2307/1970473. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [25] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models,, Oxford Lecture Series in Mathematics and its Applications, (1998). Google Scholar [26] N. Masmoudi and F. Rousset, Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations,, preprint, (). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] J. Nash, Le problème de Cauchy pour les équations différentielles d'un fluide général,, Bull. Soc. Math. France, 90 (1962), 487. Google Scholar [31] C.-L.-M.-H. Navier, Mémoire sur les lois du mouvement des fluides,, Mém. Acad. Roy. Sci. Inst. France, 6 (1823), 389. Google Scholar [32] M. Paddick, Stability and instability of Navier boundary layers,, Differential Integral Equations, 27 (2014), 893. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [38] P. Secchi, Well-posedness of characteristic symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 134 (1996), 155. doi: 10.1007/BF00379552. Google Scholar [39] P. Secchi, Some properties of anisotropic Sobolev spaces,, Arch. Math. (Basel), 75 (2000), 207. doi: 10.1007/s000130050494. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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