Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GLOBAL EXISTENCE FOR 3D NAVIER-STOKES EQUATIONS IN A LONG PERIODIC DOMAIN

  • Kim, Nam-Kwon (Department of Mathematics Chosun University) ;
  • Kwak, Min-Kyu (Department of Mathematics Chonnam National University)
  • Received : 2010.11.05
  • Published : 2012.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a long periodic domain. We show by a simple argument that a strong solution exists globally in time when the initial velocity in $H^1$ and the forcing function in $L^p$([0; T);$L^2$), T > 0, $2{\leq}p{\leq}+\infty$ satisfy a certain condition. This condition common appears for the global existence in thin non-periodic domains. Larger and larger initial data and forcing functions satisfy this condition as the thickness of the domain $\epsilon$ tends to zero.

Keywords

References

  1. J. D. Avrin, Large-eigenvalue global existence and regularity results for the Navier-Stokes equations, J. Differential Equations 127 (1996), no. 2, 365-390. https://doi.org/10.1006/jdeq.1996.0074
  2. P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago, 1988.
  3. H. Fujita and T. Kato, On the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 16 (1964), 269-315. https://doi.org/10.1007/BF00276188
  4. E. Hopf, Uber die Anfangswertaufgabe fur die hydrodynamischen Grudgleichungen, Math. Nachr. 4 (1951), 213-231. https://doi.org/10.1002/mana.3210040121
  5. D. Iftimie, The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier-Stokes equations, Bull. Soc. Math. France 127 (1999), no. 4, 473-517. https://doi.org/10.24033/bsmf.2358
  6. D. Iftimie and G. Raugel, Some results on the Navier-Stokes equations in thin 3D domains, J. Differential Equations 169 (2001), no. 2, 281-331. https://doi.org/10.1006/jdeq.2000.3900
  7. D. Iftimie, G. Raugel, and G. R. Sell, Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J. 56 (2007), no. 3, 1083-1156. https://doi.org/10.1512/iumj.2007.56.2834
  8. M. Kwak and N. Kim, Remark on global existence for 3D Navier-Stokes equations in Lipschitz domain, Submitted (2007).
  9. I. Kukavica and M. Ziane, Regularity of the Navier-Stokes equation in a thin periodic domain with large data, Discrete Contin. Dyn. Syst. 16 (2006), no. 1, 67-86. https://doi.org/10.3934/dcds.2006.16.67
  10. I. Kukavica and M. Ziane, On the regularity of the Navier-Stokes equation in a thin periodic domain, J. Differential Equations 234 (2007), no. 2, 485-506. https://doi.org/10.1016/j.jde.2006.11.020
  11. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), no. 1, 193-248. https://doi.org/10.1007/BF02547354
  12. S. Mongtgomery-Smith, Global regularity of the Navier-Stokes equations on thin three dimensional domains with periodic boundary conditions, Electron. J. Differential Equations 1999 (1999), no. 11, 1-19.
  13. G. Raugel and G. R. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993), no. 3, 503-568.
  14. G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Math. Sciences 143, Springer, Berlin, 2002.
  15. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS Regional Conference Series, No. 66, SIAM, Philadelphia, 1995.
  16. R. Temam and M. Ziane, Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. Differential Equations 1 (1996), no. 4, 499-546.