On the Inviscid Limit of the 3D Navier–Stokes Equations with Generalized Navier-Slip Boundary Conditions

Abstract

In this paper, we investigate the vanishing viscosity limit problem for the 3-dimensional (3D) incompressible Navier–Stokes equations in a general bounded smooth domain of R ³ with the generalized Navier-slip boundary conditions $u^{\varepsilon}\cdot n = 0,\ n\times(\omega^{\varepsilon}) = [B u^{\varepsilon}]_{\tau}\ {\rm on} \ \partial\varOmega$. Some uniform estimates on rates of convergence in C([0,T],L ²(Ω)) and C([0,T],H ¹(Ω)) of the solutions to the corresponding solutions of the ideal Euler equations with the standard slip boundary condition are obtained.

Keywords

Navier–Stokes equations / Slip boundary conditions / Inviscid limit

References

[1] Beavers G.S., Joseph D.D. Boundary conditions at a naturally permeable wall. J. Fluid Mech.. 1967, 30 197-207

[2] Beirão da Veiga H., Berselli L.C. Navier–Stokes equations: Greens matrices, vorticity direction, and regularity up to the boundary. J. Differ. Equ.. 2009, 246 2 597-628

[3] Beirão da Veiga H., Crispo F. Sharp inviscid limit results under Navier type boundary conditions. An L _p theory. J. Math. Fluid Mech.. 2010, 12 397-411

[4] Beirão da Veiga H., Crispo F. Concerning the W ^k,p-inviscid limit for 3D flows under a slip boundary condition. J. Math. Fluid Mech.. 2011, 13 117-135

[5] Beirão da Veiga H., Crispo F. The 3D inviscid limit result under slip boundary conditions. A negative answer. J. Math. Fluid Mech.. 2012, 14 55-59

[6] Beirão da Veiga H., Crispo F. A missed persistence property for the Euler equations and its effect on inviscid limits. Nonlinearity. 2012, 25 1661-1669

[7] Bellout H., Neustupa J., Penel P. On a ν continous family of strong solution to the Euler or Navier–Stokes equations with the Navier type boundary condition. Discrete Contin. Dyn. Syst.. 2010, 27 4 1353-1373

[8] Berselli L.C. Some results on the Navier–Stokes equations with Navier boundary conditions. Riv. Math. Univ. Parma (N. S.). 2010, 1 1 1-75

[9] Berselli, L.C., Spirito, S.: On the vanishing viscosity limit of 3D Navier–Stokes equations under slip boundary conditions in general domains. Commun. Math. Phys.

[10] Bourguignon J.P., Brezis H. Remarks on the Euler equation. J. Funct. Anal.. 1974, 15 341-363

[11] 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. 1998, 11 1625-1636

[12] Constantin P. Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations. Commun. Math. Phys.. 1986, 104 311-326

[13] Constantin P. On the Euler equations of incompressible fluids. Bull., New Ser., Am. Math. Soc.. 2007, 44 4 603-621

[14] Constantin P., Foias C. Navier Stokes Equation. 1988 Chicago: Univ. of Chicago Press

[15] Ebin D.G., Marsden J. Groups of diffeomorphisms and the notion of an incompressible fluid. Ann. Math.. 1970, 92 102-163

[16] Gie G.M., Kelliher J.P. Boundary layer analysis of the Navier–Stokes equations with generalized Navier boundary conditions. J. Differ. Equ.. 2012, 253 6 1862-1892

[17] Iftimie D., Planas G. Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions. Nonlinearity. 2006, 19 4 899-918

[18] Iftimie D., Sueur F. Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Anal.. 2011, 199 1 145-175

[19] Kato T. Nonstationary flows of viscous and ideal fluids in R3. J. Funct. Anal.. 1972, 9 296-305

[20] Kato T. Quasi-linear equations of evolution, with applications to partial differential equations. Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974; Dedicated to Konrad Jörgens). 1975 Berlin: Springer. 25-70

[21] Kato T. Chen S.S. Remarks on zero viscosity limit for non-stationary Navier–Stokes flows with boundary. Seminar on PDE. 1984 New York: Springer. 85-98

[22] Kato T., Lai C.Y. Nonlinear evolution equations and the Euler flow. J. Funct. Anal.. 1984, 56 1 15-28

[23] Kelliher J. Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J. Math. Anal.. 2006, 38 1 210-232

[24] Kelliher J. On Kato’s conditions for vanishing viscosity. Indiana Univ. Math. J.. 2007, 56 4 1711-1721

[25] Lions J.-L. Quelques Méthodes de Résolution des Problèmes aux Limites non Linéires. 1969 Paris: Dunod

[26] Maekawa, Y.: On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half plane. Preprint (2012)

[27] Masmoudi N. Remarks about the inviscid limit of the Navier–Stokes system. Commun. Math. Phys.. 2007, 270 3 777-788

[28] Masmoudi N., Rousset F. Uniform regularity for the Navier–Stokes equation with Navier boundary condition. Arch. Ration. Mech. Anal.. 2012, 203 529-575

[29] Mazzucato A. On the zero viscosity limit in incompressible fluids. Phys. Scr.. 2008, 132

[30] Navier C.L.M.H. Sur les lois de l’équilibre et du mouvement des corps élastiques. Mém. Acad. Sci. Inst. Fr.. 1827, 6 369

[31] Prizjev N.V., Troian S.M. Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces. J. Fluid Mech.. 2006, 554 25-46

[32] 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.. 1998, 192 2 433-461

[33] Sammartino M., Caflisch R.E. Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. II. Construction of the Navier–Stokes solution. Commun. Math. Phys.. 1998, 192 2 463-491

[34] Stokes G.G. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elestic solids. Trans. Camb. Philos. Soc.. 1845, 8 287-305

[35] Swann H.S.G. The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in R ³. Trans. Am. Math. Soc.. 1971, 157 373-397

[36] Temam R. Navier–Stokes Equations and Nonlinear Functional Analysis. 1995 2 Philadelphia: Society for Industrial and Applied Mathematics

[37] Thompson P.A., Troian S.M. A general boundary condition for liquid flow at solid surface. Nature. 1997, 389 360-362

[38] Wang X. A Kato type theorem on zero viscosity limit of Navier–Stokes flows. Indiana Univ. Math. J.. 2001, 50 223-241

[39] Wang X., Wang Y., Xin Z. Boundary layers in incompressible Navier–Stokes equations with Navier boundary conditions for the vanishing viscosity limit. Commun. Math. Sci.. 2010, 8 4 965-998

[40] Wang L., Xin Z., Zang A. Vanishing viscous limits for 3D Navier–Stokes equations with a Navier-slip boundary condition. J. Math. Fluid Mech.. 2012, 14 4 791-825

[41] Xiao Y.L., Xin Z.P. On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition. Commun. Pure Appl. Math.. 2007, LX 1027-1055

[42] Xiao Y.L., Xin Z.P. Remarks on the vanishing viscosity limit for 3D Navier–Stokes equations with a slip boundary condition. Chin. Ann. Math.. 2011, 32B 3 321-332

[43] Xiao Y., Xin Z. On 3D Lagrangian Navier–Stokes α model with a class of vorticity slip boundary conditions. J. Math. Fluid Mech.. 2012

[44] Xiao Y.L., Xin Z.P. A new boundary condition for the 3D Navier–Stokes equation and the vanishing viscosity limit. J. Math. Phys.. 2012, 53

[45] Xin Z., Yanagisawa T. Zero-viscosity limit of the linearized Navier–Stokes equations for a compressible viscous fluid in the half-plane. Commun. Pure Appl. Math.. 1999, 52 4 479-541