PARTIAL REGULARITY OF LERAY-HOPF WEAK SOLUTIONS TO THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS WITH HYPERDISSIPATION

We show that if u is a Leray-Hopf weak solution to the incompressible Navier-Stokes equations with hyperdissipation & alpha; E (1, 5 ) then there exists a set S & SUB; R3 such that u remains bounded outside of S at each 4 blow-up time, the Hausdorff dimension of S is bounded above by 5 - 4 & alpha; and its box-counting dimension is bounded by 31 (-16 & alpha;2 + 16 & alpha; + 5). Our approach is inspired by the ideas of Katz and Pavlovic & PRIME; (Geom. Funct. Anal. 12:2 (2002), 355-379).

Automated summary

This paper proves that if u is a Leray-Hopf weak solution to the incompressible Navier-Stokes equations with hyperdissipation α ∈ (1, 5), then there exists a set S ⊂ R3 such that u remains bounded outside of S at each blow-up time, the Hausdorff dimension of S is bounded above by 5 - 4α, and its box-counting dimension is bounded by 31(-16α² + 16α + 5). The approach in this paper is inspired by the ideas presented in Katz and Pavlovic (Geom. Funct. Anal. 12:2 (2002), 355-379).