We consider the Cauchy problem for the 3D Navier–Stokes equations and show that weak solutions satisfying suitable geometric conditions are smooth. The ﬁrst condition we consider requires the vorticity’s direction at the point y (near to x ) to be within a circular cone of a given small amplitude, with vertex at x and axis along the vorticity’s direction at x . The amplitude of the angle at the vertex of the cone depends mainly on the viscosity and on the size of the initial datum. Hence, the vorticity’s direction need not be (Holder) continuous to ensure regularity, as in the results proved in previous studies. Our new results show that a possible singularity deriving from non-trivial collision of strong vortex tubes is depleted if the angle formed by the tubes is small enough. Other conditions regarding the direction of the curl of the vorticity are considered: it is shown that if the direction of curl of the vorticity at point x is ‘nearly parallel’ to the same quantity at neighbouring points, then weak solutions are smooth.
|Titolo:||Some geometric constraints and the problem of global regularity for the Navier-Stokes equations|
|Anno del prodotto:||2009|
|Digital Object Identifier (DOI):||10.1088/0951-7715/22/10/013|
|Appare nelle tipologie:||1.1 Articolo in rivista|