LOWER BOUNDS ON BLOWING-UP SOLUTIONS OF THE THREE-DIMENSIONAL NAVIER-STOKES EQUATIONS IN (H) over dot(3/2), (H) over dot(5/2), AND (B) over dot(2,1)(5/2)

摘要

If u is a smooth solution of the Navier Stokes equations on R-3 with first blowup time T, we prove lower bounds for u in the Sobolev spaces (H) over dot(3/2), (H) over dot(5/2), and the Besov space (B) over dot(2,1)(5/2) with optimal rates of blowup: we prove the strong lower bounds parallel to u(t)parallel to((H) over dot3/2) >= c(T - t)(-1/2) and parallel to u(t)parallel to(5/2)((B) over dot2,1) >= c(T - t)(-1); in (H) over dot(5/2) we obtain limsup(t -> T)(-) (T -t) parallel to u(t)parallel to((H) over dot)5/2 >= c, a weaker result. The proofs involve new inequalities for the nonlinear term in Sobolev and Besov spaces, both of which are obtained using a dyadic decomposition of u.