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 $s\ge 0$ 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 $s\ge \frac{3}{4}$ 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 Hölder continuous solutions with epochs of regularity in the range $0<s<\frac{1}{3}$.

arXiv comment27 pages

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Cited by in corpus (1)

• On the conservation of helicity by weak solutions of the 3D Euler and inviscid MHD equations