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 \(0 < s < \frac{1}{3}\).