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

Abstract

Abstract The fractional Navier–Stokes equations on a periodic domain [ 0 , L ] 3 differ from their conventional counterpart by the replacement of the − ν Δ u Laplacian term by ν s A s u , where A = − Δ is the Stokes operator and ν s = ν L 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 .