Lower bounds on blowing-up solutions of the three-dimensional Navier--Stokes equations in $\dot H^{3/2}$, $\dot H^{5/2}$, and $\dot B^{5/2}_{2,1}$\ud \ud \ud

摘要

If u is a smooth solution of the Navier–Stokes equations on R\ud3 with first blowup time T, we prove lower bounds for u in the Sobolev spaces H˙ 3/2 , H˙ 5/2 , and the Besov space B˙ 5/2 2,1 , with optimal rates of blowup: we prove the strong lower bounds ku(t)kH˙ 3/2 ≥ c(T − t) −1/2 and ku(t)kB˙ 5/2 2,1 ≥ c(T − t) −1 , but in H˙ 5/2 we only obtain the weaker result lim supt→T − (T −t)ku(t)kH˙ 5/2 ≥ c. The\udproofs involve new inequalities for the nonlinear term in Sobolev and Besov spaces, both of which are obtained using a dyadic decomposition of u.