Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the embedding of the attractor generated by Navier-Stokes equations into finite dimensional spaces


Author: Mahdi Mohebbi
Journal: Proc. Amer. Math. Soc. 141 (2013), 3453-3465
MSC (2010): Primary 37L30; Secondary 54C25
DOI: https://doi.org/10.1090/S0002-9939-2013-11618-7
Published electronically: June 10, 2013
MathSciNet review: 3080168
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For 2-D Navier-Stokes equations on a $ C^2$ bounded domain $ \Omega $, a class of nonlinear homeomorphisms is constructed from the attractor of Navier-Stokes to curves in $ \mathbb{R}^N$ for sufficiently large $ N$. The construction uses an $ \varepsilon $-net on $ \Omega $ (so does not use the information ``near'' the boundary) and is more physically perceivable compared to abstract common embeddings.


References [Enhancements On Off] (What's this?)

  • 1. A. Ben-Artzi, A. Eden, C. Foiaş, and B. Nicolaenko, Hölder continuity for the inverse of Mañé's projection, J. Math. Anal. Appl. 178 (1993), no. 1, 22-29. MR 1231724 (94d:58091)
  • 2. P. Constantin and C. Foiaş, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. MR 972259 (90b:35190)
  • 3. A. Eden, C. Foiaş, B. Nicolaenko, and R. Temam, Exponential attractors for dissipative evolution equations, RAM: Research in Applied Mathematics, vol. 37, Masson, Paris, 1994. MR 1335230 (96i:34148)
  • 4. C. Foiaş and E. Olson, Finite fractal dimension and Hölder-Lipschitz parametrization, Indiana Univ. Math. J. 45 (1996), no. 3, 603-616. MR 1422098 (97m:58120)
  • 5. C. Foiaş and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $ 2$, Rend. Sem. Mat. Univ. Padova 39 (1967), 1-34. MR 0223716 (36:6764)
  • 6. C. Foiaş and R. Temam, Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Math. Comp. 43 (1984), no. 167, 117-133. MR 744927 (85f:35165)
  • 7. P. K. Friz, I. Kukavica, and J. C. Robinson, Nodal parametrisation of analytic attractors, Discrete Contin. Dynam. Systems 7 (2001), no. 3, 643-657. MR 1815771 (2002f:37133)
  • 8. P. K. Friz and J. C. Robinson, Parametrising the attractor of the two-dimensional Navier-Stokes equations with a finite number of nodal values, Phys. D 148 (2001), no. 3-4, 201-220. MR 1820361 (2002a:35023)
  • 9. G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. II, Springer Tracts in Natural Philosophy, vol. 39, Springer-Verlag, New York, 1994, Nonlinear steady problems. MR 1284206 (95i:35216b)
  • 10. -, An introduction to the Navier-Stokes initial-boundary value problem, Fundamental directions in mathematical fluid mechanics, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2000, pp. 1-70. MR 1798753 (2002c:35207)
  • 11. G. P. Galdi and S. Rionero, A priori estimates, continuous dependence and stability for solutions to Navier-Stokes equations on exterior domains, Riv. Mat. Univ. Parma (4) 5 (1979), part 2, 533-556 (1980). MR 584228 (81j:35093)
  • 12. E. Hopf, A mathematical example displaying features of turbulence, Communications on Appl. Math. 1 (1948), 303-322. MR 0030113 (10:716a)
  • 13. B. R. Hunt and V. Yu. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity 12 (1999), no. 5, 1263-1275. MR 1710097 (2001a:28009)
  • 14. R. J. Knops and L. E. Payne, On the stability of solutions of the Navier-Stokes equations backward in time, Arch. Rational Mech. Anal. 29 (1968), 331-335. MR 0226222 (37:1812)
  • 15. O. A. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, Journal of Mathematical Sciences 3 (1975), 458-479.
  • 16. R. Mañé, On the dimension of the compact invariant sets of certain nonlinear maps, Dynamical systems and turbulence, Warwick, 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, Springer, Berlin, 1981, pp. 230-242. MR 654892 (84k:58119)
  • 17. J. C. Robinson, Infinite-dimensional dynamical systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001, An introduction to dissipative parabolic PDEs and the theory of global attractors. MR 1881888 (2003f:37001a)
  • 18. -, Attractors and finite-dimensional behaviour in the Navier-Stokes equations, Instructional conference on mathematical analysis of hydrodynamics (Edinburgh), ICMS, June 2003.
  • 19. -, A topological delay embedding theorem for infinite-dimensional dynamical systems, Nonlinearity 18 (2005), no. 5, 2135-2143. MR 2164735 (2006i:37166)
  • 20. R. Temam, Navier-Stokes equations. Theory and numerical analysis, North-Holland Publishing Co., Amsterdam, 1977, Studies in Mathematics and its Applications, Vol. 2. MR 0609732 (58:29439)
  • 21. -, Infinite-dimensional dynamical systems in mechanics and physics, second ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR 1441312 (98b:58056)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37L30, 54C25

Retrieve articles in all journals with MSC (2010): 37L30, 54C25


Additional Information

Mahdi Mohebbi
Affiliation: Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, Pennsylvania 15261
Email: mam175@pitt.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11618-7
Received by editor(s): May 24, 2011
Received by editor(s) in revised form: October 3, 2011, and December 8, 2011
Published electronically: June 10, 2013
Additional Notes: This work was partially supported by NSF grant DMS-1062381.
Communicated by: Walter Craig
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society