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 −νΔu Laplacian term by νsAsu , where A=−Δ is the Stokes operator and νs=νL2(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=13 ; s=34 ; s=56 and s=54 . In particular: (i) for s>13 we prove an analogue of one of the Prodi–Serrin regularity criteria; (ii) for s⩾34 we find an equation of local energy balance and; (iii) for s>56 we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi–Serrin criterion for s>13 suggests the sharpness of the construction using convex integration of Hölder continuous solutions with epochs of regularity in the range 0<s<13 .