Vorticity directions near the blow-up time for the 3DNavier-Stokes flows with infinite energy
Hideyuki Miura (Osaka University)
This talk is based on a joint work with Yoshikazu Giga (University of Tokyo). We
consider the incompressible Navier-Stokes equations in R3 :{∂u/∂t − ∆u + u · ∇u + ∇p = 0 in R3 × (0, T ),(NS)div u = 0 in R3 × (0, T ).
There is a large number of work on regularity criterion or non blow up criterion forthe 3D Navier-Stokes ﬂow started by Serrin [Se]. Most of them are analytic in the sensethat boundedness for some (scale-invariant) quantity is assumed. In 1993, Constantin andFeﬀerman [CF] gave a geometric condition for the vorticity direction ζ = ω/|ω|, whereω = curl u. It says that if the vorticity direction is Lipschitz continuous in space uniformly
in time in the region where vorticity is large, then the Leray-Hopf weak solution is regular.
Since then there are several improvement, e.g., [BB, CKL, GZ]. However, all these work
discuss a solution having a ﬁnite energy. In this talk, we consider a mild solution (i.e.,a solution of integral equations) of (NS) in R3 × (0, T ), where a solution is just boundedin space and may not have a ﬁnite energy. We shall give a condition for the vorticity
direction so that the solution can be extended smoothly across t = T . Roughly speaking
our result reads: the solution does not blow up at t = T if the blow up is type I and thevorticity direction is just uniformly continuous in space variables. For the assumption on
the vorticity direction, our result improves existing one. However, we are forced to assume
the growth of L∞ -norm of solution is bounded by a self-similar rate which is called type I
blow up rate, that is,||u||L∞ (R3 ) (t) ≤ C0 (T − t)1/2for t ∈ (0, T ),(I)

where the time t = T is considered as a possible blow up time. We now state our main
result.
Theorem Let u be a mild solution of (NS) in R3 × (0, T ) satisfying (I). For a given d > 0let η be a modulus of continuity such that|ζ(x, t) − ζ(y, t)| ≤ η(|x − y|) for all x, y ∈ Ωd (t), t ∈ (0, T )(CA)

where Ωd (t) = {x ∈ R3 | |ω(x, t)| > d}. Then u does not blow up at t = T .
Remark In [CF] η is taken η(σ) = Aσ in (CA) where A is a positive constant. In [BB] ηis taken η(σ) = Aσ 1/2 . However, the authors did not assume that the singularity is type I.
Our theorem is proved by a blow up analysis. Assume the contrary so that the solution
blows up at t = T . We are able to rescale the solution near the blow up time by Mk :=∥u∥∞ (tk ) ↑ +∞ for some {tk }∞k=1 with tk ↑ T . Taking a blow up limit of the sequenceof the rescaled solutions we obtain a bounded backward global solution of (NS) u deﬁned

in R3 × (−∞, 0). It follows from the assumption (I) that the vorticity of the bounded
backward global solution ω = rot u is not identically zero. On the other hand, we also show
the following Liouville type theorem, which contradicts with the above conclusion.
Proposition Assume that u is a mild solution of (N S) in R3 × (0, T ) which is boundedin R3 × (0, T − δ) for all δ > 0. If ζ satisfies (CA), then ω ≡ 0.
The key observation of the proof is that the assumption (CA) implies the blow up limit u
is a 2D Navier-Stokes ﬂow. Since the vorticity of the 2D Navier-Stokes ﬂow satisﬁes thestrong maximum principle, we are able to show that the vorticity must be zero.
All previous results [CF, BB, GZ, CKL] are proved by integral estimates where somespecial structure of the nonlinear term plays a crucial role, while our result is by the blowup argument and the Liouville type theorem. The blow up argument is not only simple
but also clariﬁes the nature of our problem. We note that similar blow up argument iseﬀectively applied to the analysis of various kind of equations, e.g., [G, H, PQS].
We also remark that the question whether the type I blow up solutions for (NS) exist or
not is still open. Necas, Ruzicka and Sverak [NRS] proved that non-trivial backward selfsimilar solutions do not exist. Giga, Inui and Matsui [GIM] showed that the solution does
not blow up if C0 is suﬃciently small. Recently, Koch, Nadirashvili, Seregin and Sverak[KNSS] proved that type I axisymmetric ﬂows must be regular by using a Liouville typetheorem for (NS); see also [CSYT] for a diﬀerent proof. Our result gives another suﬃcientcondition for nonexistence of the type I blow up solutions.

References[BB] Beirao da Veiga, H., Berselli, L. C.: Diﬀerential Integral Equations 15 (2002), 345-356.
[CKL] Chae, D., Kang, K., Lee, J.: Comm. PDE. 32 (2007), 1189-1207.
[CSYT] Chen, C.-C., Strain, R. M., Yau, H.-T., Tsai, T.-P.: IMRN (2008), rnn016.
[CF] Constantin, P., Feﬀerman, C.: Indiana Univ. Math. J. 42 (1993), 775-789.
[G] Giga, Y.: Comm. Math. Phys. 103 (1986), 415-421.
[GIM] Giga, Y., Inui, K., Matsui, S.: Quad. Mat., 4 (1999), 27-68.
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[KNSS] Koch, G., Nadirashvili, N., Seregin, G., Sverak, V.:Acta. Math. 203 (2009), 83-105.
[NRS] Necas, J., Ruzicka, M., Sverak, V.: Acta Math. 176 (1996), 283-294.
[PQS] Polácik, P., Quittner, P., Souplet, P.: Indiana Univ. Math. J. 56 (2007), 879-908.
[Se] Serrin, J.: Nonlinear Problems, R. E. Langer (eds) (1963), 69-98.