Lower bounds on blow up solutions of the three-dimensional Navier–Stokes equations in homogeneous Sobolev spaces

Suppose that u(t) is a solution of the three-dimensional Navier–Stokes equations, either on the whole space or with periodic boundary conditions, that has a singularity at time T. In this paper we show that the norm of u(T − t) in the homogeneous Sobolev space [12pt]{minimal}{document}Ḣˢ{document}Ḣs must be bounded below by cst−(2s−1)/4 for 1/2 < s < 5/2 (s ≠ 3/2), where cs is an absolute constant depending only on s; and by [12pt]{minimal}{document}cₛ u₀ L²(5-2s)/5t-2s/5{document}cs‖u0‖L2(5−2s)/5t−2s/5 for s > 5/2. (The result for 1/2 < s < 3/2 follows from well-known lower bounds on blowup in Lp spaces.) We show in particular that the local existence time in [12pt]{minimal}{document}Ḣˢ( R³){document}Ḣs(R3) depends only on the [12pt]{minimal}{document}Ḣˢ{document}Ḣs-norm for 1/2 < s < 5/2, s ≠ 3/2.