Monday 04 Nov 2013: Remarks on the regularity for the Navier-Stokes equations: self-similarity and criticality revisited (Taylor and Francis sponsored)
Koji Ohkitani - University of Sheffield
Harrison 170 16:00-17:00
We consider the regularity issues of the Navier-Stokes equations
in the whole space, centering on self-similarity and criticality
(scale-invariance). It is well-known that energy is critical in 2D,
enstrophy in 4D and a "helicity-like integral" in 3D.
By using the critical conditions, we first give shortened proofs of absence
of self-similar blowup, i.e. of the fact that the Leray equations have trivial
solutions only: Necas, Ruzicka and Sverak (1996), Tsai(1998).
We then consider a non-steady version of the Leray equations after applying
dynamic scaling transformations. We show how the long-time asymptotics
of their solutions can be consistent with absence of self-similar
blowup. Implications of the analysis are discussed.