Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

HOMOGENIZATION OF THE NON-STATIONARY STOKES EQUATIONS WITH PERIODIC VISCOSITY

  • Published : 2009.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

We study the periodic homogenization of the non-stationary Stokes equations. The fundamental homogenization theorem and corrector theorem are proved under a very general assumption on the viscosity coefficients and data. The proofs are based on a weak formulation suitable for an application of classical Tartar's method of oscillating test functions. Such a weak formulation is derived by adapting an argument in Teman's book [Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1984].

Keywords

References

  1. A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Applied Mathematical Sciences 151, Springer-Verlag, Berlin, 2002
  2. A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978
  3. M. E. Bogovskii, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR 248 (1979), no. 5, 1037.1040. (Russian); English Transl.: Soviet Math Dokl. 20 (1979), 1094.1098
  4. S. Brahim-Stsmane, G. A. Francfort, and F. Murat, Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl. (9) 71 (1992), no. 3, 197.231
  5. D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications, 17. The Clarendon Press, Oxford University Press, New York, 1999
  6. C. Conca, On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures Appl. (9) 64 (1985), no. 1, 31.75
  7. G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994
  8. F. W. Gehring, The Lp-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265.277 https://doi.org/10.1007/BF02392268
  9. M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math. 330 (1982), 173.214
  10. N. G. Meyers, An Lpe-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 189.206
  11. J. Necas and I. Hlavacek, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Studies in Applied Mechanics 3, Elsevier Scientific Publishing Co., Amsterdam-New York, 1980
  12. E. Sanchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Phys. 127, Springer-Verlag, Berlin, 1980
  13. R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1984
  14. L. Wang, On Korn's inequality, J. Comput. Math. 21 (2003), no. 3, 321.324

Cited by

  1. Periodic homogenization of the non-stationary Navier–Stokes type equations vol.28, pp.3-4, 2017, https://doi.org/10.1007/s13370-016-0463-7