• Previous Article
    Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source
  • NHM Home
  • This Issue
  • Next Article
    A link between microscopic and macroscopic models of self-organized aggregation
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 & 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, 22 (1990). Google Scholar

[2]

Lamberto Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes,, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308. Google Scholar

[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. doi: 10.4208/cicp.110909.160310s. Google Scholar

[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. doi: 10.1090/S0025-5718-2011-02469-5. Google Scholar

[5]

Philippe G. Ciarlet, "An Introduction to Differential Geometry with Applications to Elasticity,", Springer, 78/79 (2005). Google Scholar

[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. doi: 10.1016/S0021-7824(99)00032-X. Google Scholar

[7]

Weinan E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation,, Acta Math. Sin. (Engl. Ser.), 16 (2000), 207. doi: 10.1007/s101140000034. Google Scholar

[8]

Wiktor Eckhaus, Boundary layers in linear elliptic singular perturbation problems,, SIAM Rev., 14 (1972), 225. Google Scholar

[9]

Gung-Min Gie, Singular perturbation problems in a general smooth domain,, Asymptot. Anal., 62 (2009), 227. Google Scholar

[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. doi: 10.1080/00036810903437796. Google Scholar

[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. doi: 10.3934/dcds.2010.26.1213. Google Scholar

[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. doi: 10.1016/j.jde.2012.06.008. Google Scholar

[13]

Emmanuel Grenier and Olivier Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems,, J. Differential Equations, 143 (1998), 110. doi: 10.1006/jdeq.1997.3364. Google Scholar

[14]

Makram Hamouda and Roger Temam, Some singular perturbation problems related to the Navier-Stokes equations,, in, (2007), 197. doi: 10.1142/9789812770493_0011. Google Scholar

[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. Google Scholar

[16]

Mark H. Holmes, "Introduction to Perturbation Methods,", Texts in Applied Mathematics 20, 20 (1995). doi: 10.1007/978-1-4612-5347-1. Google Scholar

[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. doi: 10.1007/s00205-010-0320-z. Google Scholar

[18]

Wilhelm Klingenberg, "A Course in Differential Geometry,", Graduate Texts in Mathematics 51, 51 (1978). Google Scholar

[19]

J.-L. Lions, "Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal,", Lecture Notes in Mathematics 323, 323 (1973). Google Scholar

[20]

Nader Masmoudi, The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary,, Arch. Rational Mech. Anal., 142 (1998), 375. doi: 10.1007/s002050050097. Google Scholar

[21]

Robert E. O'Malley, Jr., "Singular Perturbation Analysis for Ordinary Differential Equations,", Communications of the Mathematical Institute, 5 (1977). Google Scholar

[22]

Madalina Petcu, Euler equation in a 3D channel with a noncharacteristic boundary,, Differential Integral Equations, 19 (2006), 297. Google Scholar

[23]

Shagi-Di Shih and R. Bruce Kellogg, Asymptotic analysis of a singular perturbation problem,, SIAM J. Math. Anal., 18 (1987), 1467. doi: 10.1137/0518107. Google Scholar

[24]

R. Temam, Behaviour at time $t=0$ of the solutions of semilinear evolution equations,, J. Differential Equations, 43 (1982), 73. Google Scholar

[25]

R. Temam and X. Wang, Remarks on the Prandtl equation for a permeable wall,, ZAMM Z. Angew. Math. Mech., 80 (2000), 835. doi: 10.1002/1521-4001(200011)80:11/12<835::AID-ZAMM835>3.3.CO;2-0. Google Scholar

[26]

R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case,, J. Differential Equations, 179 (2002), 647. doi: 10.1006/jdeq.2001.4038. Google Scholar

[27]

Roger Temam, On the Euler equations of incompressible perfect fluids,, J. Functional Analysis, 20 (1975), 32. Google Scholar

[28]

Roger Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", AMS Chelsea Publishing, (2001). Google Scholar

[29]

Roger Temam and Xiao Ming Wang, Asymptotic analysis of the linearized Navier-Stokes equations in a channel,, Differential Integral Equations, 8 (1995), 1591. Google Scholar

[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. Google Scholar

[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), 807. Google Scholar

[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. Google Scholar

[33]

Xiaoming Wang, Examples of boundary layers associated with the incompressible Navier-Stokes equations,, Chin. Ann. Math. Ser. B, 31 (2010), 781. doi: 10.1007/s11401-010-0597-0. Google Scholar

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, 22 (1990). Google Scholar

[2]

Lamberto Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes,, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308. Google Scholar

[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. doi: 10.4208/cicp.110909.160310s. Google Scholar

[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. doi: 10.1090/S0025-5718-2011-02469-5. Google Scholar

[5]

Philippe G. Ciarlet, "An Introduction to Differential Geometry with Applications to Elasticity,", Springer, 78/79 (2005). Google Scholar

[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. doi: 10.1016/S0021-7824(99)00032-X. Google Scholar

[7]

Weinan E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation,, Acta Math. Sin. (Engl. Ser.), 16 (2000), 207. doi: 10.1007/s101140000034. Google Scholar

[8]

Wiktor Eckhaus, Boundary layers in linear elliptic singular perturbation problems,, SIAM Rev., 14 (1972), 225. Google Scholar

[9]

Gung-Min Gie, Singular perturbation problems in a general smooth domain,, Asymptot. Anal., 62 (2009), 227. Google Scholar

[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. doi: 10.1080/00036810903437796. Google Scholar

[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. doi: 10.3934/dcds.2010.26.1213. Google Scholar

[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. doi: 10.1016/j.jde.2012.06.008. Google Scholar

[13]

Emmanuel Grenier and Olivier Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems,, J. Differential Equations, 143 (1998), 110. doi: 10.1006/jdeq.1997.3364. Google Scholar

[14]

Makram Hamouda and Roger Temam, Some singular perturbation problems related to the Navier-Stokes equations,, in, (2007), 197. doi: 10.1142/9789812770493_0011. Google Scholar

[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. Google Scholar

[16]

Mark H. Holmes, "Introduction to Perturbation Methods,", Texts in Applied Mathematics 20, 20 (1995). doi: 10.1007/978-1-4612-5347-1. Google Scholar

[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. doi: 10.1007/s00205-010-0320-z. Google Scholar

[18]

Wilhelm Klingenberg, "A Course in Differential Geometry,", Graduate Texts in Mathematics 51, 51 (1978). Google Scholar

[19]

J.-L. Lions, "Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal,", Lecture Notes in Mathematics 323, 323 (1973). Google Scholar

[20]

Nader Masmoudi, The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary,, Arch. Rational Mech. Anal., 142 (1998), 375. doi: 10.1007/s002050050097. Google Scholar

[21]

Robert E. O'Malley, Jr., "Singular Perturbation Analysis for Ordinary Differential Equations,", Communications of the Mathematical Institute, 5 (1977). Google Scholar

[22]

Madalina Petcu, Euler equation in a 3D channel with a noncharacteristic boundary,, Differential Integral Equations, 19 (2006), 297. Google Scholar

[23]

Shagi-Di Shih and R. Bruce Kellogg, Asymptotic analysis of a singular perturbation problem,, SIAM J. Math. Anal., 18 (1987), 1467. doi: 10.1137/0518107. Google Scholar

[24]

R. Temam, Behaviour at time $t=0$ of the solutions of semilinear evolution equations,, J. Differential Equations, 43 (1982), 73. Google Scholar

[25]

R. Temam and X. Wang, Remarks on the Prandtl equation for a permeable wall,, ZAMM Z. Angew. Math. Mech., 80 (2000), 835. doi: 10.1002/1521-4001(200011)80:11/12<835::AID-ZAMM835>3.3.CO;2-0. Google Scholar

[26]

R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case,, J. Differential Equations, 179 (2002), 647. doi: 10.1006/jdeq.2001.4038. Google Scholar

[27]

Roger Temam, On the Euler equations of incompressible perfect fluids,, J. Functional Analysis, 20 (1975), 32. Google Scholar

[28]

Roger Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", AMS Chelsea Publishing, (2001). Google Scholar

[29]

Roger Temam and Xiao Ming Wang, Asymptotic analysis of the linearized Navier-Stokes equations in a channel,, Differential Integral Equations, 8 (1995), 1591. Google Scholar

[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. Google Scholar

[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), 807. Google Scholar

[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. Google Scholar

[33]

Xiaoming Wang, Examples of boundary layers associated with the incompressible Navier-Stokes equations,, Chin. Ann. Math. Ser. B, 31 (2010), 781. doi: 10.1007/s11401-010-0597-0. Google Scholar

[1]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595

[2]

Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277

[3]

José Luiz Boldrini, Luís H. de Miranda, Gabriela Planas. On singular Navier-Stokes equations and irreversible phase transitions. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2055-2078. doi: 10.3934/cpaa.2012.11.2055

[4]

Gung-Min Gie, Makram Hamouda, Roger Témam. Boundary layers in smooth curvilinear domains: Parabolic problems. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1213-1240. doi: 10.3934/dcds.2010.26.1213

[5]

Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113

[6]

Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769

[7]

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355

[8]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[9]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[10]

Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353

[11]

Linjie Xiong. Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary. Kinetic & Related Models, 2018, 11 (3) : 469-490. doi: 10.3934/krm.2018021

[12]

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

[13]

Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153

[14]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349

[15]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433

[16]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747

[17]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403

[18]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319

[19]

Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717

[20]

Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]