Note on Leray's criterion for Navier–Stokes singularity. (August 2022) - Tran, Chuong V.
Abstract: Leray's criterion for Navier–Stokes singularity states that if a solution to the Navier–Stokes equations becomes singular at t = T ∗, then it is necessary that the velocity norm | | u | | L s ( R 3 ), for s > 3, grows unbounded in the manner | | u | | L s ( R 3 ) ≥ C s / ( T ∗ − t ) ( s − 3 ) / 2 s as t → T ∗ . Here C s may depend on s . This note examines Leray's necessary condition for singularity in some detail. Optimality of the exponent ( s − 3 ) / 2 s is confirmed and constraints on C s are derived and discussed. Furthermore, assuming singular growth of | | u | | L s ( R 3 ) exactly respecting Leray's scaling, a logarithmic constraint on the divergence of | | u | | L 3 ( R 3 ) is deduced.