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

Abstract

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 $s$ 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, in the fractional setting we prove an analogue of one of the Prodi-Serrin regularity criteria ($s > \frac{1}{3}$), an equation of local energy balance ($s \geq \frac{3}{4}$) and an infinite hierarchy of weak solution time averages ($s > \frac{5}{6}$). The existence of our analogue of the Prodi-Serrin criterion for $s > \frac{1}{3}$ suggests that the convex integration schemes that construct H\"older-continuous solutions with epochs of regularity for $s < \frac{1}{3}$ are sharp with respect to the value of $s$