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December  2012, 7(4): 741-766. doi: 10.3934/nhm.2012.7.741

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, United States, United States, United States

Received  January 2012 Revised  May 2012 Published  December 2012

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.
Citation: Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Networks and Heterogeneous Media, 2012, 7 (4) : 741-766. doi: 10.3934/nhm.2012.7.741
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.

show all references

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.

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