The fractional Navier–Stokes equations on a periodic domain [0, L]
3 differ from
their conventional counterpart by the replacement of the −ν∆u Laplacian
term by νsA
su, where A = −∆ is the Stokes operator and νs = νL
2(s−1)
is the
viscosity parameter. Four critical values of the exponent s ⩾ 0 have been identified where functional properties of solutions of the fractional Navier–Stokes
equations change. These values are: s =
1
3
; s =
3
4
; s =
5
6
and s =
5
4
. In particular: (i) for s >
1
3 we prove an analogue of one of the Prodi–Serrin regularity
criteria; (ii) for s ⩾
3
4 we find an equation of local energy balance and; (iii)
for s >
5
6 we find an infinite hierarchy of weak solution time averages. The
existence of our analogue of the Prodi–Serrin criterion for s >
1
3
suggests the
sharpness of the construction using convex integration of Hölder continuous
solutions with epochs of regularity in the range 0 < s <
1
3
.