December  2017, 12(4): 525-550. doi: 10.3934/nhm.2017022

Homogenization of stokes system using bloch waves

1. 

Centre de Mathématiques Appliquées, Ecole Polytechnique, CNRS, Université Paris-Saclay, Palaiseau, France

2. 

Centre for Applicable Matematics, Tata Institute of Fundamental Research, Bangalore, India

3. 

Current address: Mathematics Department, IIT-Bombay, Mumbai, India

Received  September 2016 Revised  April 2017 Published  October 2017

In this work, we study the Bloch wave homogenization for the Stokes system with periodic viscosity coefficient. In particular, we obtain the spectral interpretation of the homogenized tensor. The presence of the incompressibility constraint in the model raises new issues linking the homogenized tensor and the Bloch spectral data. The main difficulty is a lack of smoothness for the bottom of the Bloch spectrum, a phenomenon which is not present in the case of the elasticity system. This issue is solved in the present work, completing the homogenization process of the Stokes system via the Bloch wave method.

Citation: Grégoire Allaire, Tuhin Ghosh, Muthusamy Vanninathan. Homogenization of stokes system using bloch waves. Networks & Heterogeneous Media, 2017, 12 (4) : 525-550. doi: 10.3934/nhm.2017022
References:
[1]

G. Allaire and M. Briane, Multiscale convergence and reiterated homogenisation, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 297-342.  doi: 10.1017/S0308210500022757.  Google Scholar

[2]

G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Engrg., 187 (2000), 91-117.  doi: 10.1016/S0045-7825(99)00112-7.  Google Scholar

[3]

G. Allaire and Y. Capdeboscq, Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface, Ann. Mat. Pura Appl. (4), 181 (2002), 247-282.  doi: 10.1007/s102310100040.  Google Scholar

[4]

G. AllaireY. Capdeboscq and A. Piatnitski, Homogenization and localization with an interface, Indiana Univ. Math. J., 52 (2003), 1413-1446.  doi: 10.1512/iumj.2003.52.2352.  Google Scholar

[5]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[6]

G. Allaire, Shape Optimization by the Homogenization Method volume 146 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  Google Scholar

[7]

G. AllaireC. ConcaL. Friz and J.~H. Ortega, On Bloch waves for the Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 1-28 (electronic).   Google Scholar

[8]

A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, volume 5 of Studies in Mathematics and its Applications, North-Holland Publishing Co. , Amsterdam-New York, 1978.  Google Scholar

[9]

Y. Capdeboscq and M.S. Vogelius, Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements, M2AN Math. Model. Numer. Anal., 37 (2003), 227-240.  doi: 10.1051/m2an:2003024.  Google Scholar

[10]

H. J. Choe and H. Kim, Homogenization of the non-stationary {S}tokes equations with periodic viscosity, J. Korean Math. Soc., 46 (2009), 1041-1069.  doi: 10.4134/JKMS.2009.46.5.1041.  Google Scholar

[11]

C. Conca, J. Planchard and M. Vanninathan, Fluids and Periodic Structures, volume 38 of RAM: Research in Applied Mathematics, John Wiley & Sons, Ltd. , Chichester; Masson, Paris, 1995.  Google Scholar

[12]

C. ConcaR. Orive and M. Vanninathan, Bloch approximation in homogenization and applications, SIAM J. Math. Anal., 33 (2002), 1166-1198 (electronic).  doi: 10.1137/S0036141001382200.  Google Scholar

[13]

C. Conca and M. Vanninathan, Homogenization of periodic structures via {B}loch decomposition, SIAM J. Appl. Math., 57 (1997), 1639-1659.  doi: 10.1137/S0036139995294743.  Google Scholar

[14]

S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators, Asymptot. Anal., 39 (2004), 15-44.   Google Scholar

[15]

V. Girault and P. -A. Raviart, Finite Element Methods for Navier-Stokes Equations, volume 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

[16]

U. Hornung, Homogenization and Porous Media, Springer, New York, 1997. doi: 10.1007/978-1-4612-1920-0.  Google Scholar

[17]

R. Morgan and I. Babuska, An approach for constructing families of homogenized equations for periodic media, SIAM J. Math. Anal., 22 (1991), 16-33.  doi: 10.1137/0522002.  Google Scholar

[18]

J.H. Ortega and E. Zuazua, Generic simplicity of the eigenvalues of the Stokes system in two space dimensions, Adv. Differential Equations, 6 (2001), 987-1023.   Google Scholar

[19]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, New York, 1978.  Google Scholar

[20]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, volume 129 of Springer Lecture Notes in Physics, Springer-Verlag, Berlin, 1980.  Google Scholar

[21]

S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of linear elasticity system, ESAIM Control Optim. Calc. Var., 11 (2005), 542-573 (electronic).  doi: 10.1051/cocv:2005018.  Google Scholar

[22]

C. Wilcox, Theory of bloch waves, J. Anal. Math., 33 (1978), 146-167.  doi: 10.1007/BF02790171.  Google Scholar

show all references

References:
[1]

G. Allaire and M. Briane, Multiscale convergence and reiterated homogenisation, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 297-342.  doi: 10.1017/S0308210500022757.  Google Scholar

[2]

G. Allaire and Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Engrg., 187 (2000), 91-117.  doi: 10.1016/S0045-7825(99)00112-7.  Google Scholar

[3]

G. Allaire and Y. Capdeboscq, Homogenization and localization for a 1-D eigenvalue problem in a periodic medium with an interface, Ann. Mat. Pura Appl. (4), 181 (2002), 247-282.  doi: 10.1007/s102310100040.  Google Scholar

[4]

G. AllaireY. Capdeboscq and A. Piatnitski, Homogenization and localization with an interface, Indiana Univ. Math. J., 52 (2003), 1413-1446.  doi: 10.1512/iumj.2003.52.2352.  Google Scholar

[5]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[6]

G. Allaire, Shape Optimization by the Homogenization Method volume 146 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  Google Scholar

[7]

G. AllaireC. ConcaL. Friz and J.~H. Ortega, On Bloch waves for the Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 1-28 (electronic).   Google Scholar

[8]

A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, volume 5 of Studies in Mathematics and its Applications, North-Holland Publishing Co. , Amsterdam-New York, 1978.  Google Scholar

[9]

Y. Capdeboscq and M.S. Vogelius, Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements, M2AN Math. Model. Numer. Anal., 37 (2003), 227-240.  doi: 10.1051/m2an:2003024.  Google Scholar

[10]

H. J. Choe and H. Kim, Homogenization of the non-stationary {S}tokes equations with periodic viscosity, J. Korean Math. Soc., 46 (2009), 1041-1069.  doi: 10.4134/JKMS.2009.46.5.1041.  Google Scholar

[11]

C. Conca, J. Planchard and M. Vanninathan, Fluids and Periodic Structures, volume 38 of RAM: Research in Applied Mathematics, John Wiley & Sons, Ltd. , Chichester; Masson, Paris, 1995.  Google Scholar

[12]

C. ConcaR. Orive and M. Vanninathan, Bloch approximation in homogenization and applications, SIAM J. Math. Anal., 33 (2002), 1166-1198 (electronic).  doi: 10.1137/S0036141001382200.  Google Scholar

[13]

C. Conca and M. Vanninathan, Homogenization of periodic structures via {B}loch decomposition, SIAM J. Appl. Math., 57 (1997), 1639-1659.  doi: 10.1137/S0036139995294743.  Google Scholar

[14]

S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators, Asymptot. Anal., 39 (2004), 15-44.   Google Scholar

[15]

V. Girault and P. -A. Raviart, Finite Element Methods for Navier-Stokes Equations, volume 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

[16]

U. Hornung, Homogenization and Porous Media, Springer, New York, 1997. doi: 10.1007/978-1-4612-1920-0.  Google Scholar

[17]

R. Morgan and I. Babuska, An approach for constructing families of homogenized equations for periodic media, SIAM J. Math. Anal., 22 (1991), 16-33.  doi: 10.1137/0522002.  Google Scholar

[18]

J.H. Ortega and E. Zuazua, Generic simplicity of the eigenvalues of the Stokes system in two space dimensions, Adv. Differential Equations, 6 (2001), 987-1023.   Google Scholar

[19]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, New York, 1978.  Google Scholar

[20]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, volume 129 of Springer Lecture Notes in Physics, Springer-Verlag, Berlin, 1980.  Google Scholar

[21]

S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of linear elasticity system, ESAIM Control Optim. Calc. Var., 11 (2005), 542-573 (electronic).  doi: 10.1051/cocv:2005018.  Google Scholar

[22]

C. Wilcox, Theory of bloch waves, J. Anal. Math., 33 (1978), 146-167.  doi: 10.1007/BF02790171.  Google Scholar

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