Abstract
We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier–Stokes equation with the Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in L ∞. This allows us to obtain the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bardos C.: Existence et unicité de la solution de l’équation d’Euler en dimension deux. J. Math. Anal. Appl. 40, 769–790 (1972)
Bardos C., Rauch J.: Maximal positive boundary value problems as limits of singular perturbation problems. Trans. Am. Math. Soc. 270(2), 377–408 (1982)
Basson A., Gérard-Varet D.: Wall laws for fluid flows at a boundary with random roughness. Commun. Pure Appl. Math. 61(7), 941–987 (2008)
Beirão da Veiga H.: Vorticity and regularity for flows under the Navier boundary condition. Commun. Pure Appl. Anal. 5(4), 907–918 (2006)
Beirão da Veiga H., Crispo F.: Concerning the W k,p-inviscid limit for 3-d flows under a slip boundary condition. J. Math. Fluid Mech. 13, 117–135 (2011)
Clopeau T., Mikelić A., Robert R.: On the vanishing viscosity limit for the 2D incompressible Navier–Stokes equations with the friction type boundary conditions. Nonlinearity 11(6), 1625–1636 (1998)
Gérard-Varet D., Masmoudi N.: Relevance of the slip condition for fluid flows near an irregular boundary. Commun. Math. Phys. 295(1), 99–137 (2010)
Gisclon M., Serre D.: Étude des conditions aux limites pour un système strictement hyberbolique via l’approximation parabolique. C. R. Acad. Sci. Paris Sér. I Math. 319(4), 377–382 (1994)
Grenier E., Guès O.: Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Differ. Equ. 143(1), 110–146 (1998)
Grenier E., Masmoudi N.: Ekman layers of rotating fluids, the case of well prepared initial data. Comm. Partial Differ. Equ. 22(5–6), 953–975 (1997)
Grenier E., Rousset F.: Stability of one-dimensional boundary layers by using Green’s functions. Commun. Pure Appl. Math. 54(11), 1343–1385 (2001)
Guès O.: Problème mixte hyperbolique quasi-linéaire caractéristique. Commun. Partial Differ. Equ. 15(5), 595–645 (1990)
Hörmander L.: Pseudo-differential operators and non-elliptic boundary problems. Ann. Math. 83(2), 129–209 (1966)
Iftimie D., Planas G.: Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions. Nonlinearity 19(4), 899–918 (2006)
Iftimie D., Sueur F.: Viscous boundary layers for the Navier–Stokes equations with the navier slip conditions. Arch. Rational Mech. Anal. 199, 145–175 (2011)
Jang J., Masmoudi N.: Well-posedness for compressible Euler equations with physical vacuum singularity. Commun. Pure Appl. Math. 62(10), 1327–1385 (2009)
Kato T.: Nonstationary flows of viscous and ideal fluids in R 3. J. Funct. Anal. 9, 296–305 (1972)
Kato, T.: Remarks on the Euler and Navier–Stokes equations in R 2. In: Nonlinear Functional Analysis and its Applications, Part 2 (Berkeley, Calif., 1983). Amer. Math. Soc., Providence, R.I., pp. 1–7, 1986
Kelliher J.P.: Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J. Math. Anal. 38(1), 210–232 (2006)
Klainerman S., Majda A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34(4), 481–524 (1981)
Lions, P.-L.: Mathematical Topics in Fluid Mechanics, Vol. 1. The Clarendon Press, Oxford University Press, New York. Incompressible Models. Oxford Science Publications, 1996
Masmoudi N.: The Euler limit of the Navier–Stokes equations, and rotating fluids with boundary. Arch. Rational Mech. Anal. 142(4), 375–394 (1998)
Masmoudi N.: Ekman layers of rotating fluids: the case of general initial data. Commun. Pure Appl. Math. 53(4), 432–483 (2000)
Masmoudi N.: Examples of singular limits in hydrodynamics. In: Evolutionary Equations, Vol. III. Handb. Differ. Equ. Elsevier/North-Holland, Amsterdam, 195–276, 2007
Masmoudi N.: Remarks about the inviscid limit of the Navier–Stokes system. Commun. Math. Phys. 270(3), 777–788 (2007)
Masmoudi N., Rousset F.: Stability of oscillating boundary layers in rotating fluids. Ann. Sci. Éc. Norm. Supér. (4) 41(6), 955–1002 (2008)
Masmoudi N., Saint-Raymond L.: From the Boltzmann equation to the Stokes-Fourier system in a bounded domain. Commun. Pure Appl. Math. 56(9), 1263–1293 (2003)
Métivier G., Schochet S.: The incompressible limit of the non-isentropic Euler equations. Arch. Rational Mech. Anal. 158(1), 61–90 (2001)
Métivier, G., Zumbrun, K.: Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Am. Math. Soc. 175, 826, vi+107 (2005)
Rousset F.: Stability of large Ekman boundary layers in rotating fluids. Arch. Rational Mech. Anal. 172(2), 213–245 (2004)
Rousset F.: Characteristic boundary layers in real vanishing viscosity limits. J. Differ. Equ. 210(1), 25–64 (2005)
Saint-Raymond L.: Weak compactness methods for singular penalization problems with boundary layers. SIAM J. Math. Anal. 41(1), 153–177 (2009)
Sammartino M., Caflisch R.E.: Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space, I. Existence for Euler and Prandtl equations. Commun. Math. Phys. 192(2), 433–461 (1998)
Swann H.S.G.: The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in R 3. Trans. Am. Math. Soc. 157, 373–397 (1971)
Tartakoff D.S.: Regularity of solutions to boundary value problems for first order systems. Indiana Univ. Math. J. 21, 1113–1129 (1972)
Temam R.: On the Euler equations of incompressible perfect fluids. J. Funct. Anal. 20(1), 32–43 (1975)
Temam R., Wang X.: Boundary layers associated with incompressible Navier–Stokes equations: the noncharacteristic boundary case. J. Differ. Equ. 179(2), 647–686 (2002)
Xiao Y., Xin Z.: On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition. Commun. Pure Appl. Math. 60(7), 1027–1055 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Dafermos
Rights and permissions
About this article
Cite this article
Masmoudi, N., Rousset, F. Uniform Regularity for the Navier–Stokes Equation with Navier Boundary Condition. Arch Rational Mech Anal 203, 529–575 (2012). https://doi.org/10.1007/s00205-011-0456-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-011-0456-5