Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 6 (2001) > Paper 5 open journal systems 


Statistically Stationary Solutions to the 3D Navier-Stokes Equations do not show Singularities

Franco Flandoli, Università di Pisa
Marco Romito, Università di Firenze


Abstract
If $mu$ is a probability measure on the set of suitable weak solutions of the 3D Navier-Stokes equations, invariant for the time-shift, with finite mean dissipation rate, then at every time t the set of singular points is empty $mu$-a.s. The existence of a measure $mu$ with the previous properties is also proved; it may describe a turbulent asymptotic regime.


Full text: PDF

Pages: 1-15

Published on: August 17, 2001
Bibliography
  1. Bell, J. B. and Marcus, D. L. (1992), Vorticity intensification and the transition to turbulence in the three-dimensional Euler equation, Comm. Math. Phys. 147, 371-394 MR 93c:76048
  2. Caffarelli, L. Kohn, R. and Nirenberg, L. (1982), Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. XXXV, 771-831 MR 84m:35097
  3. Chorin, A. J. (1982), The evolution of a turbulent vortex, Comm. Math. Phys. 83, 517-535. MR 83g:76042
  4. Chorin A. J. (1994), Vorticity and Turbulence, Springer-Verlag, New York. MR 95m:76043
  5. Flandoli, F. and Schmalfuss, B. (1999), Weak solutions and attractors for the 3 dimensional Navier-Stokes equations with non-regular force, J. Dyn. Diff. Eq. 11, Nr. 2, 355-398. MR 2000j:60076
  6. Fursikov, A. V. (1983), Statistical extremal problems and unique solvability of the three dimensional Navier-Stokes system under almost all initial conditions, PMM USSR 46, Nr. 5, 637-644. MR 84k:76046
  7. Gallavotti, G. (1996), Ipotesi per una introduzione alla meccanica dei fluidi, Quaderni CNR-GNFM, Roma.
  8. Grauer, R. and Sideris, T. (1991), Numerical computation of 3D incompressible ideal fluids with swirl, Phys. Rev. Lett. 67, 3511-3514.
  9. Kerr, R. (1993), Evidence for a singularity of the three dimensional incompressible Euler equation, Phys. Fluids A 6, 1725-1739. MR 94d:76015
  10. Lanford III, O. E. (1975), Time evolution of large classical systems, in: Dynamical systems, theory and applications, Lecture Notes in Physics, Vol. 38, Springer-Verlag, Berlin. MR 57#18653
  11. Lin, F. (1998), A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math. LI, 241-257. MR 98k:35151
  12. Lions, P. L. (1996), Mathematical Topics in Fluid Mechanics, Vol. I, Clarendon Press, Oxford. 98b:76001
  13. Romito, M. (2000), Ph.D. Thesis, Pisa.
  14. Scheffer, V. (1980), The Navier-Stokes equations on a bounded domain, Comm. Math. Phys. 73, 1-42. MR 81f:35097
  15. Sell, G. (1996), Global attractor for the 3D Navier-Stokes equations, J. Dyn. Diff. Eq. 8. MR 98e:35127
  16. Serrin, J. (1962), On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal. 9, 187-195. MR 25:346
  17. Siegmund-Schultze, R. (1985), On non-equilibrium dynamics of multidimensional infinite particle systems in the translation invariant case, Comm. Math. Phys. 100, 245-265. MR 87a:82014
  18. Sohr, H. and von Wahl, W. (1986), On the regularity of the pressure of weak solutions of Navier-Stokes equations, Arch. Math. (Basel) 46, 428-439. MR 87g:35190
  19. Temam, R. (1977), The Navier-Stokes equations, North Holland. MR 58:29439
  20. Temam, R. (1983), Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia. MR 86f:35152
  21. Temam, R. (1988), Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York. MR 89m:58056
  22. Vishik M. I. and Fursikov, A. V. (1980), Mathematical Problems of Statistical Hydromechanics, Kluwer, Dordrecht. MR 82g:35095
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489