FM 92-1

Author: Giovanni Gallavotti

Title: Some rigorous results about 3d Navier-Stokes.

Abstract:

There are very few rigorous results on 3D Navier-Stokes equations.
Here I select and review a few among the most remarkable:
\item{1.} Leray's local theorem
\item{2.} Scheffer's local theorem
\item{3.} Caffarelli, Kohn, Nirenberg regularity theorem.

I provide an interpretation of the scaling ideas in 1, 2, 3 with
particular attention to the similarity of 3) with the theory of
relevant variables in the renormalization group methods. The
connection of 3) with Kolmogorov- Obuchov theory (quoted as (KO) below)
is attempted.

Finally I present a comment inspired by Chorin's ideas on the
vorticity filaments dynamics (3D). To simplify a subject, quite
intricate in itself, only a fluid in a periodic container $\O$
is considered.

[Text of two lectures delivered at the Les Houches NATO-ASI meeting
``Turbulence in spatially ordered systems'', january, 1992.]

Keywords: Fluid dynamics, Renormalization group, Kolmogorov Obuchov
theory, vorticity filaments, Hausdorff dimension, Non linear PDE's.


Fisica, Universita' di Roma La Sapienza,
P.le Moro 2, 00185, Roma, Italia.
e-mail giovanni@ipparco.roma1.infn.it
tel. 6-49914370, fax 6-4957697

Home page: http://www.math.rutgers.edu/~giovanni