Lower Bounds on Blowing-Up Solutions of the Three-Dimensional Navier--Stokes Equations in $\dot H^{3/2}$, $\dot H^{5/2}$, and $\dot B^{5/2}_{2,1}$

Abstract

Abstract. 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 spacesḢ 3/2 ,Ḣ 5/2 , and the Besov spaceḂ 5/2 2,1 , with optimal rates of blowup: we prove the strong lower bounds u(t) Ḣ3/2 ≥ c(T − t) −1/2 and u(t) Ḃ 5/2 2,1 ≥ c(T − t) −1 ; inḢ 5/2 we obtain lim sup t→T − (T − t) u(t) Ḣ5/2 ≥ 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.

Persistence time of solutions of the three-dimensional Navier-Stokes equations in Sobolev-Gevrey classes

“…It is worth mentioning that the differential inequalities for the evolution of the Gevrey norms that one obtains in these cases are non-autonomous; estimates of existence times of these given in Lemma 4.4 and Lemma 5.3, though elementary, may be new as well. Moreover, in Corollary 2.2, we give an alternate proof for the persistence in the Sobolev class H s for the entire range 1 2 < s < 5 2 , thus unifying the results in [49] and [16,19,44] and showing that the case 3 2 is not a borderline in our approach. Furthermore, unlike in [16,44], our method is elementary and avoids any harmonic analysis machinery such as paraproducts and Littlewood-Paley decomposition.…”

“…Moreover, in Corollary 2.2, we give an alternate proof for the persistence in the Sobolev class H s for the entire range 1 2 < s < 5 2 , thus unifying the results in [49] and [16,19,44] and showing that the case 3 2 is not a borderline in our approach. Furthermore, unlike in [16,44], our method is elementary and avoids any harmonic analysis machinery such as paraproducts and Littlewood-Paley decomposition.…”

“…Recently, several authors [2,16,18,19,44,49] have obtained "optimal" existence times, and the associated blow-up rates, assuming they exist, for solutions of the 3D NSE in Sobolev spaces H s , s > 1 2 . In particular, in [49], by employing a scaling argument, Robinson, Sadowski and Silva established that the optimal existence time of a (strong) solution of the NSE in the whole space R 3 , for initial data in H s , s > 1 2 , is necessarily given by T (u 0 ) 1…”

“…Evidently, the existence time is optimal for 1 2 < s < 5 2 , s = 3 2 , while the existence time for s > 5 2 , though not optimal, is the best known to-date. The borderline cases, namely s = 3 2 , s = 5 2 , were subsequently considered by varying methods in [16,18,19,44], including Littlewodd-Paley decomposition and other harmonic analysis tools, the upshot being that the optimal existence time T ∼ 1 u0 2 H s also holds for s = 3 2 , while the optimal existence time in H 5/2 is still open.…”

Local well-posedness and blowup rates for the Navier-Stokes equations in subcritical Lorentz spaces

“…By standard Sobolev embeddings, Leray's result (1.7) implies similar results in a certain range of subcritical homogeneous Sobolev spaces. We note here that there has also been much recent interest in extending the subcritical Sobolev range for such results (see for example [1,2,4,5,6,10,15,16,20]).…”