Phase transitions in the fractional three-dimensional Navier-Stokes equations

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 $0 < s < \frac{1}{3}$.