Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary

Gung-Min Gie; Makram Hamouda; Roger Temam; Gung-Min Gie; Makram Hamouda; Roger Temam

doi:10.3934/nhm.2012.7.741

Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 741-766. doi: 10.3934/nhm.2012.7.741

Previous Article Next Article

Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary

1.
The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, Indiana 47405

Received: 01 January 2012 Revised: 01 May 2012
35B25, 35C20, 35K05, 76D07, 76D10.

We consider the Navier-Stokes equations of an incompressible fluid in a three dimensional curved domain with permeable walls in the limit of small viscosity. Using a curvilinear coordinate system, adapted to the boundary, we construct a corrector function at order $ε^j$, $j=0,1$, where $ε$ is the (small) viscosity parameter. This allows us to obtain an asymptotic expansion of the Navier-Stokes solution at order $ε^j$, $j=0,1$, for $ε$ small . Using the asymptotic expansion, we prove that the Navier-Stokes solutions converge, as the viscosity parameter tends to zero, to the corresponding Euler solution in the natural energy norm. This work generalizes earlier results in [14] or [26], which discussed the case of a channel domain, while here the domain is curved.
- Boundary layers,
- singular perturbations,
- Navier-Stokes equations,
- curvilinear coordinates
Citation: Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary[J]. Networks and Heterogeneous Media, 2012, 7(4): 741-766. doi: 10.3934/nhm.2012.7.741

Related Papers:

Abstract

We consider the Navier-Stokes equations of an incompressible fluid in a three dimensional curved domain with permeable walls in the limit of small viscosity. Using a curvilinear coordinate system, adapted to the boundary, we construct a corrector function at order $ε^j$, $j=0,1$, where $ε$ is the (small) viscosity parameter. This allows us to obtain an asymptotic expansion of the Navier-Stokes solution at order $ε^j$, $j=0,1$, for $ε$ small . Using the asymptotic expansion, we prove that the Navier-Stokes solutions converge, as the viscosity parameter tends to zero, to the corresponding Euler solution in the natural energy norm. This work generalizes earlier results in [14] or [26], which discussed the case of a channel domain, while here the domain is curved.

References

[1]	S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, "Boundary Value Problems in Mechanics of Nonhomogeneous Fluids," Studies in Mathematics and its Applications 22, North-Holland Publishing Co., Amsterdam, 1990. Translated from the Russian.
[2]	Lamberto Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.
[3]	Qingshan Chen, Zhen Qin and Roger Temam, Numerical resolution near $t=0$ of nonlinear evolution equations in the presence of corner singularities in space dimension 1, Commun. Comput. Phys., 9 (2011), 568-586. doi: 10.4208/cicp.110909.160310s
[4]	Qingshan Chen, Zhen Qin and Roger Temam, Treatment of incompatible initial and boundary data for parabolic equations in higher dimension, Math. Comp., 80 (2011), 2071-2096. doi: 10.1090/S0025-5718-2011-02469-5
[5]	Philippe G. Ciarlet, "An Introduction to Differential Geometry with Applications to Elasticity," Springer, Dordrecht, 2005, Reprinted from J. Elasticity 78/79, 2005.
[6]	B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. (9), 78 (1999), 461-471. doi: 10.1016/S0021-7824(99)00032-X
[7]	Weinan E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.), 16 (2000), 207-218. doi: 10.1007/s101140000034
[8]	Wiktor Eckhaus, Boundary layers in linear elliptic singular perturbation problems, SIAM Rev., 14 (1972), 225-270.
[9]	Gung-Min Gie, Singular perturbation problems in a general smooth domain, Asymptot. Anal., 62 (2009), 227-249.
[10]	Gung-Min Gie, Makram Hamouda and Roger Temam, Asymptotic analysis of the Stokes problem on general bounded domains: the case of a characteristic boundary, Appl. Anal., 89 (2010), 49-66. doi: 10.1080/00036810903437796
[11]	Gung-Min Gie, Makram Hamouda and Roger Temam, Boundary layers in smooth curvilinear domains: Parabolic problems, Discrete Contin. Dyn. Syst.-A, 26 (2010), 1213-1240. doi: 10.3934/dcds.2010.26.1213
[12]	Gung-Min Gie and James P. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Differential Equations, 253 (2012), 1862-1892. doi: 10.1016/j.jde.2012.06.008
[13]	Emmanuel Grenier and Olivier Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems, J. Differential Equations, 143 (1998), 110-146. doi: 10.1006/jdeq.1997.3364
[14]	Makram Hamouda and Roger Temam, Some singular perturbation problems related to the Navier-Stokes equations, in "Advances in Deterministic and Stochastic Analysis," World Sci. Publ., Hackensack, NJ, (2007), 197-227. doi: 10.1142/9789812770493_0011
[15]	Makram Hamouda and Roger Temam, Boundary layers for the Navier-Stokes equations. The case of a characteristic boundary, Georgian Math. J., 15 (2008), 517-530.
[16]	Mark H. Holmes, "Introduction to Perturbation Methods," Texts in Applied Mathematics 20, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-5347-1
[17]	Dragoş Iftimie and Franck 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
[18]	Wilhelm Klingenberg, "A Course in Differential Geometry," Graduate Texts in Mathematics 51, Springer-Verlag, New York, 1978, Translated from the German by David Hoffman.
[19]	J.-L. Lions, "Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal," Lecture Notes in Mathematics 323, Springer-Verlag, Berlin, 1973.
[20]	Nader Masmoudi, The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary, Arch. Rational Mech. Anal., 142 (1998), 375-394. doi: 10.1007/s002050050097
[21]	Robert E. O'Malley, Jr., "Singular Perturbation Analysis for Ordinary Differential Equations," Communications of the Mathematical Institute, Rijksuniversiteit Utrecht 5, Rijksuniversiteit Utrecht Mathematical Institute, Utrecht, 1977.
[22]	Madalina Petcu, Euler equation in a 3D channel with a noncharacteristic boundary, Differential Integral Equations, 19 (2006), 297-326.
[23]	Shagi-Di Shih and R. Bruce Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal., 18 (1987), 1467-1511. doi: 10.1137/0518107
[24]	R. Temam, Behaviour at time $t=0$ of the solutions of semilinear evolution equations, J. Differential Equations, 43 (1982), 73-92.
[25]	R. Temam and X. Wang, Remarks on the Prandtl equation for a permeable wall, ZAMM Z. Angew. Math. Mech., 80 (2000), 835-843. Special Issue on the Occasion of the 125th Anniversary of the Birth of Ludwig Prandtl. doi: 10.1002/1521-4001(200011)80:11/12<835::AID-ZAMM835>3.3.CO;2-0
[26]	R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case, J. Differential Equations, 179 (2002), 647-686. doi: 10.1006/jdeq.2001.4038
[27]	Roger Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis, 20 (1975), 32-43.
[28]	Roger Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1984 edition.
[29]	Roger Temam and Xiao Ming Wang, Asymptotic analysis of the linearized Navier-Stokes equations in a channel, Differential Integral Equations, 8 (1995), 1591-1618.
[30]	Roger Temam and Xiaoming Wang, Asymptotic analysis of the linearized Navier-Stokes equations in a general $2$D domain, Asymptot. Anal., 14 (1997), 293-321.
[31]	Roger Temam and Xiaoming Wang, On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25 (1997) (1998), 807-828. Dedicated to Ennio De Giorgi.
[32]	M. I. Višik and L. A. Ljusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Amer. Math. Soc. Transl. (2), 20 (1962), 239-364.
[33]	Xiaoming Wang, Examples of boundary layers associated with the incompressible Navier-Stokes equations, Chin. Ann. Math. Ser. B, 31 (2010), 781-792. doi: 10.1007/s11401-010-0597-0

Reader Comments

Your name:*

Email:*
© 2012 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)