45 research outputs found
An anisotropic regularity condition for the 3D incompressible Navier-Stokes equations for the entire exponent range
We show that a suitable weak solution to the incompressible Navier-Stokes
equations on is regular on
if belongs to for any and ,
which is a logarithmic-type variation of a Morrey space in time. For each
this space is, up to a logarithm, critical with respect to the
scaling of the equations, and contains all spaces that are subcritical, that is for which .Comment: 10 page
Phase transitions in the fractional three-dimensional Navier-Stokes equations
The fractional Navier-Stokes equations on a periodic domain
differ from their conventional counterpart by the replacement of the
Laplacian term by , where is the Stokes operator and is the viscosity
parameter. Four critical values of the exponent have been identified where
functional properties of solutions of the fractional Navier-Stokes equations
change. These values are: ; ; and
. In particular, in the fractional setting we prove an analogue
of one of the Prodi-Serrin regularity criteria (), an equation
of local energy balance () and an infinite hierarchy of
weak solution time averages (). The existence of our analogue
of the Prodi-Serrin criterion for suggests that the convex
integration schemes that construct H\"older-continuous solutions with epochs of
regularity for are sharp with respect to the value of