The fractional Navier-Stokes equations on a periodic domain $[0,\,L]^{ 3}$ differ from their conventional counterpart by the replacement of the $-\nu\Delta\mathbf{ u}$ Laplacian term by $\nu_{ s}A^{ s}\mathbf{ u}$, where $A= - \Delta$ is the Stokes operator and $\nu_{ s} = \nu L^{ 2(s-1)}$ is the viscosity parameter. Four critical values of the exponent $s\geq 0$ have been identified where functional properties of solutions of the fractional Navier-Stokes equations change. These values are: $s=\frac{ 1}{ 3}$; $s=\frac{ 3}{ 4}$; $s=\frac{ 5}{ 6}$ and $s=\frac{ 5}{ 4}$. In particular: i) for $s > \frac{ 1}{ 3}$ we prove an analogue of one of the Prodi-Serrin regularity criteria; ii) for $s \geq \frac{ 3}{ 4}$ we find an equation of local energy balance and; iii) for $s > \frac{ 5}{ 6}$ we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi-Serrin criterion for $s > \frac{ 1}{ 3}$ suggests the sharpness of the construction using convex integration of H\"older continuous solutions with epochs of regularity in the range $0 < s < \frac{ 1}{ 3}$.
Categories: math.AP nlin.CD physics.flu-dyn