Global regularity for solutions of the three dimensional Navier-Stokes equation with almost two dimensional initial data

In this paper, we will prove a new result that guarantees the global existence of solutions to the Navier-Stokes equation in three dimensions when the initial data is sufficiently close to being two dimensional. This result interpolates between the global existence of smooth solutions for the two dimensional Navier-Stokes equation with arbitrarily large initial data, and the global existence of smooth solutions for the Navier-Stokes equation in three dimensions with small initial data in $\dot{H}^\frac{1}{2}$. This result states that the closer the initial data is to being two dimensional, the larger the initial data can be in $\dot{H}^\frac{1}{2}$ while still guaranteeing the global existence of smooth solutions. In the whole space, this set of almost two dimensional initial data is unbounded in the critical space $\dot{H}^\frac{1}{2},$ but is bounded in the critical Besov spaces $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}$ for all $2<p\leq +\infty.$ On the torus, however, this approach does give examples of arbitrarily large initial data in the endpoint Besov space $\dot{B}^{-1}_{\infty,\infty}$ that generate global smooth solutions to the Navier-Stokes equation. In addition to these new results, we will also sharpen the constants in a number of previously known estimates for the growth of solutions to the Navier-Stokes equation.


Introduction
The Navier-Stokes equation, which governs viscous, incompressible flow, is one of the most fundamental equations in fluid dynamics. The incompressible Navier-Stokes equations with no external forces is given by where u ∈ R 3 denotes the velocity, p the pressure, and ν > 0 is the viscosity. The pressure is completely determined in terms of u, by taking the divergence of both sides of the equation, which yields We refer here to the Navier-Stokes equation, rather than the Navier-Stokes equations, because this PDE is best viewed not as a system of equations, but as an evolution equation on the space of divergence free vector fields.
Two other objects which play a crucial role in Navier-Stokes analysis are the vorticity and the strain, which represent the anti-symmetric and symmetric parts of ∇u respectively. The vorticity is given by taking the curl of the velocity, ω = ∇ × u, while the strain is the matrix given by S ij = 1 2 ∂u j ∂x i + ∂u i ∂x j . The evolution equation for vorticity is given by The evolution equation for the strain is given by S t + (u · ∇)S − ∆S + S 2 + 1 4 ω ⊗ ω − 1 4 |ω| 2 I 3 + Hess(p) = 0. (1.4) Before we proceed further we should define a number of spaces. For all s ∈ R, H s R 3 will be the Hilbert space with norm and for all − 3 2 < s < 3 2 ,Ḣ s R 3 will be the homogeneous Hilbert space with norm Note that when referring to H s R 3 ,Ḣ s R 3 , orL p R 3 , the R 3 will often be omitted for brevity's sake. All Hilbert and Lebesgue norms are taken over R 3 unless otherwise specified. Finally we will define the subspace of divergence free vector fields inside each of these spaces. For all − 3 2 < s < 3 2 , defineḢ s df ⊂Ḣ s R 3 ; R 3 bẏ H s df = u ∈Ḣ s R 3 ; R 3 : ξ ·û(ξ) = 0, almost everywhere ξ ∈ R 3 . (

1.8)
For all 1 ≤ q ≤ +∞, define L q df ⊂ L q R 3 ; R 3 by L q df = u ∈ L q R 3 ; R 3 for all f ∈ C ∞ c R 3 , u, ∇f = 0. (1.9) Note that this definition makes sense, because in f ∈ H s or f ∈Ḣ s implies thatf (ξ) is well defined almost everywhere. We will also note that becauseḢ 0 = L 2 , so we have two different definitions of L 2 df . This is not a problem as both definitions are equivalent. The standard notion of weak solutions to PDEs corresponds to integrating against test functions. Leray first proved the existence of weak solutions satisfying a certain energy inequality [21]. Leray showed that for all initial data u 0 ∈ L 2 R 3 , ∇ · u 0 = 0 in the sense of distributions, there exists a weak solution u ∈ L ∞ [0, +∞); L 2 ∩ L 2 [0, +∞);Ḣ 1 to the Navier-Stokes equations in the sense of integrating against smooth test functions, satisfying the energy inequality, for all t > 0. This energy inequality holds with equality for smooth solutions to the Navier-Stokes equations, but a weak solution in u ∈ L ∞ [0, +∞); L 2 ∩ L 2 [0, +∞);Ḣ 1 does not have enough regularity for us to integrate by parts to conclude that (u · ∇)u, u = 0, which is what is needed to prove that the energy equality holds.
Definition 1.2. Let u be a solution to the Navier-Stokes equation, then the energy is given by K(t) = 1 2 u(·, t) 2 L 2 and the enstrophy is given by E(t) = 1 2 ω(·, t) 2 L 2 . Note that the energy and enstrophy can be alternatively defined in terms of norms on u, ω, or S. This is because of the following isometry proved by the author in [26]. (1.11) While the global existence of Leray solutions to the Navier-Stokes equations is well established, the global existence of smooth solutions remains a major open problem. A notion of solution better adapted to the Navier-Stokes regularity problem is the notion of mild solutions introduced by Kato and Fujita in [11]. Before defining mild solutions, we will define the Leray projection. Proposition 1.4 (Helmholtz decomposition). Suppose 1 < q < +∞. For all v ∈ L q (R 3 ; R 3 ) there exists a unique u ∈ L q (R 3 ; R 3 ), ∇ · u = 0 and ∇f ∈ L q (R 3 ; R 3 ) such that v = u + ∇f. Note because we do not have any assumptions of higher regularity, we will say that ∇ · u = 0, if for all u · ∇φ = 0, (1.12) and we will say that ∇f is a gradient if for all w ∈ C ∞ c (R 3 ; R 3 ), ∇ · w = 0, we have Furthermore there exists B q ≥ 1 depending only on q, such that ||u|| L q ≤ B q ||v|| L q , (1.14) and ||∇f || L q ≤ B q ||v|| L q .
(1.16) Definition 1.5 (Mild solutions). Suppose u ∈ C [0, T ];Ḣ 1 df . Then u is a mild solution to the Navier-Stokes equation if u(·, t) = e νt∆ u 0 + t 0 e ν(t−τ )∆ P df ((−u · ∇)u) (·, τ ) dτ, (1.17) where e t∆ is the heat operator given by convolution with the heat kernel; that is to say, e t∆ u 0 is the solution of the heat equation after time t, with initial data u 0 .
Fujita and Kato proved the local existence of mild solutions for initial data inḢ s df , s > 1 2 in [11]. This was extended to intial data in L q df , q > 3 by Kato in [16]. In the case where s = 1, their result is the following. The argument is based on a fixed point theorem, as a map associated with Definition 1.5 is a contraction mapping for sufficiently small times. These arguments, however, cannot guarantee the existence of a smooth solutions for arbitrarily large times. When discussing regularity for the Navier-Stokes equation it is useful to define T max , the maximal time of existence for a smooth solution corresponding to some initial data. Definition 1.7. For all u 0 ∈Ḣ 1 df , if there is a mild solution of the Navier-Stokes equation u ∈ C [0, +∞);Ḣ 1 df , u(·, 0) = u 0 , then T max = +∞. If there is not a mild solution globally in time with initial data u 0 , then let T max < +∞ be the time such that u ∈ C [0, T max );Ḣ 1 df , u(·, 0) = u 0 , is a mild solution to the Navier-Stokes equation that cannot be extended beyond T max . That is, for all T > T max there is no mild solution u ∈ C [0, T );Ḣ 1 df , u(·, 0) = u 0 .
It remains one of the biggest open questions in nonlinear PDEs, indeed one of the Millennium Problems put forward by the Clay Institute, whether the Navier-Stokes equation have smooth solutions globally in time [10]. Note in particular that the Clay Millenium problem can be equivalently stated in terms of Definition 1.7 as: show T max = +∞ for all u 0 ∈ H 1 df or provide a counterexample. It is known that the Navier-Stokes equation must have global smooth solutions for small initial data in certain scale-critical function spaces. In particular, Fujita and Kato also proved in [11] the global existence of smooth solutions to the Navier-Stokes equation for small initial data inḢ 1 2 df . Theorem 1.8. There exists C > 0 such that for all u 0 ∈Ḣ This result was then extended to L 3 by Kato [16] and to BM O −1 by Koch and Tataru [17]. We will note here that the Navier-Stokes equation is invariant under the rescaling u λ (x, t) = λu(λx, λ 2 t), and therefore u 0 generates a global smooth solution if and only if, u 0,λ (x) = λu 0 (λx) generates a global smooth solution for all λ > 0. It is easy to check that each of these norms are invariant with respect to this rescaling of the initial data.
The main theorem of this paper establishes a new result guaranteeing the existence of global smooth solutions for initial data that are arbitrarily large inḢ 1 2 , so long two components of the vorticity are sufficiently small in the critical Hilbert space.
u 0 generates a unique, global smooth solution to the Navier-Stokes equation u ∈ C (0, +∞); H 1 df , that is T max = +∞. Note that the smallness condition can be equivalently stated as Very little is known in general about the existence of smooth solutions globally in time with arbitrarily large initial data. Ladyzhenskaya proved the existence of global smooth solutions for swirl-free axisymmetric initial data [19], which gives a whole family of arbitrarily large initial data with globally smooth solutions. Mahalov, Titi, and Leibovich showed global regularity for solutions with a helical symmetry in [25]. In light of the Koch-Tataru theorem guaranteeing global regularity for small initial data in BM O −1 , it has been an active area of research to find examples of solutions that are large in BM O −1 that generate global smooth solutions, or even stronger, to find initial data large in B −1 ∞,∞ ⊃ BM O −1 , which is the maximal scale invariant space. Because both swirl free, axisymetric vector fields and helically symmetric vector fields form subspaces of divergence free vector fields, clearly these are examples of initial data large in B −1 ∞,∞ . Gallagher and Chemin showed the existence of initial data that generate global smooth solutions that are large in B −1 ∞,∞ on the torus by taking highly oscillatory initial data [5]. More recently Kukavica, Rusin, and Ziane exhibited a class of non-oscillatory initial data, large in B −1 ∞,∞ , that generate global smooth solutions [18].
Because B −1 ∞,∞ is the largest scale invariant space, this space is the correct way to measure the size some class of initial data. In order to have a genuine large data global regularity result, it is necessary to show that the set of initial data generating global smooth solutions is unbounded in B −1 ∞,∞ . Unfortunately, while the set of initial data satisfying the hypotheses of Theorem 1.9 is unbounded inḢ 1 2 , it is bounded in a whole family of scale critical Besov spaces.
df be the set of almost two dimensional initial data satisfying the hypothesis of Theorem 1.9: Then Γ 2d is unbounded inḢ Note here that for all 2 ≤ p ≤ +∞,Ḃ −1+ 3 p p,∞ R 3 is invariant under the re-scaling u 0,λ (x) = λu 0 (λx), the rescaling that preserves the solution set of the Navier-Stokes equation. Remark 1.11. A version of Theorem 1.9 holds on the torus as well. The statement is essentially the same, with the only difference being the value of the constants, because the Sobolev embedding may have different sharp constants on the torus. Interestingly, on the torus the set of almost two dimensional initial data for which we can prove global regularity is unbounded inḂ −1 ∞,∞ T 3 , which is not the case on the whole space. This allows us to provide examples of large initial data on the torus that generate global smooth solutions to the Navier-Stokes equations. We will discuss this in more detail in section 5. In particular, Theorem 5.7 is the analogous result on the torus to the whole space result Theorem 1.9.
Unlike the three dimensional case, there are global smooth solutions to the Navier-Stokes equation in two dimensions. This is because in two dimensions the energy equality is scale critical, while in three dimensions the energy inequality is supercritical. This is also because vortex stretching occurs in three dimensions, but not in two dimensions, so the enstrophy is decreasing for solutions of the two dimensional Navier-Stokes equations. Given that the Navier-Stokes equation has global smooth solutions in two dimensions, one natural approach to the extending small data regularity results to arbitrarily large initial data, would be to show global regularity for the solutions that are, in some sense, approximately two dimensional.
There are also a number of previous results guaranteeing global regularity for solutions three dimensional solutions of the Navier Stokes equations with almost two dimensional initial data. One approach to almost two dimensional initial data on the torus is to consider three dimensional initial data that is a perturbation of two dimensional initial data. Note that this approach is available on the torus, because L 2 df T 2 forms a subspace of L 2 df T 3 , so we can consider perturbations of this subspace. It is not, however, available on the whole space, as nonzero vector fields in L 2 df R 2 , lose integrability when extended to three dimensions, and so L 2 df R 2 does not define a subspace of L 2 df (R 3 ). Iftimie proved that small perturbations of two dimensional initial data must have smooth solutions to the Navier-Stokes equation globally in time. Another approach is based on re-scaling, to make the the initial data vary slowly in one direction. This approach was used by Gallagher and Chemin in [6] and extended by Gallagher, Chemin, and Paicu in [7] and by Paicu and Zhang in [27]. We will prove global regularity based on rescaling the vorticity, rather than the velocity, as this rescaling operates better with the divergence free constraint. The result we will prove is the following.
and define u 0,ǫ using the Biot-Savart law by For all u 0 ∈ H 1 df and for all 0 < a <  (1.23) In section 2, we will discuss previous regularity results and estimates for enstrophy growth, and we will sharpen some of the constants involved in these estimates. In section 3, we will consider the evolution of ω h and prove Theorem 1.9. In section 4, we will consider the question of boundedness in Besov spaces, proving Theorem 1.10. In section 5, we will state the results mentioned in the paragraph above precisely and prove Theorem 1.12. We will then discuss the relationship between these previous results and Theorem 1.9 and Theorem 1.12 in detail.

Small Data Results
We will begin by recalling an identity for enstrophy growth proven by Chae in the context of smooth solutions of the Euler equation [3] and independently by the author using different methods [26]. Recalling the isometry in Proposition 1.3, we will consider enstrophy in terms of the S(·, t) 2 L 2 .
be a mild solution to the Navier-Stokes equation, and let S = ∇ sym u, then for all 0 < t < T max As an immediate corollary, we have the following result proved by the author in [26] that follows directly from Proposition 2.1 and the condition tr(S) = 0.
be a mild solution to the Navier-Stokes equation, and let S = ∇ sym u, then for all 0 < t < T max Using Proposition 2.2 and the fractional Sobolev inequality we will be able to prove a cubic differential inequality for the growth of enstrophy. The sharp fractional Sobolev inequality was first proven by Lieb [22]. . Then for all f ∈Ḣ − 1 2 R 3 ,

3)
and for all f We will note in particular that the two inequalities in Lemma 2.3 are dual to each other because L 3 and L 3 2 are dual spaces, andḢ 1 2 andḢ − 1 2 are dual spaces, which is why the two ineqaulities have the same sharp constant. For more references on this inequality see also chapter 4 in [23] and the summary of these results in [8]. We can now prove a cubic differential inequality for the growth of enstrophy. Then for all 0 < t < T max , we have E ′ (t) ≤ 1 3,456π 4 ν 3 E(t) 3 . Furthermore, if u ∈ C [0, T max ); H 1 df , then for all 0 < t < T max , we have K ′ (t) = −2νE(t).
Proof. The equality K ′ (t) = −2νE(t) is the classic energy equality for smooth solutions of the Navier-Stokes equations first proven by Leray [21]. We will now prove the first inequality. We begin with the estimate for enstrophy growth in Corollary 2.2: Next we apply the fractional Sobolev inequality in Lemma 2.3, and observe Interpolating between L 2 andḢ 1 and simplifying the constant, we find that Substituting r = S Ḣ1 , we find Computing the derivative we find that f ′ (r) = −4νr + 3Br L 2 and that f attains its global maximum at r 0 , we conclude that This completes the proof.
The cubic bound on the growth of enstrophy is not new, however a closer analysis of the strain allows a major improvement in the constant. The best known estimate [1,24,28] for enstrophy growth that does not make use of the identity for enstrophy growth in terms of the determinant of strain in Proposition 2.1 is (2.14) The author then improved the constant in this inequality significantly; using Proposition 2.1, the author proved in [26] a cubic differential inequality controlling the growth of enstrophy, in the case where ν = 1, although there is no loss of generality in the proof: the proof in the case of ν > 0 is entirely analogous. The proof in [26] relied on the sharp Sobolev inequality proven by Talenti [29], which we will state below.
. Then for all f ∈ L 6 R 3

16)
and for all f ∈ L As in the fractional Sobolev inequality, we will note in particular that the two inequalities in Lemma 2.5 are dual to each other because L 6 and L 6 5 are dual spaces, andḢ 1 andḢ −1 are dual spaces, which is why the constant in both inequalities is the same.
In [26], the author first interpolated between L 2 and L 6 and then applied Lemma 2.5, showing It is possible to obtain a sharper constant by first applying the fractional Sobolev inequality and then interpolating between L 2 andḢ 1 . Proceeding this way, we conclude 2 , using the fractional Sobolev inequality results in a sharper bound on enstrophy growth.
Using the bounds in Proposition 2.4, we will be able to prove a small data global existence result in terms of the product of energy and enstrophy.
That is, there exists a unique, smooth solution to the Navier Stokes equation u ∈ C [0, +∞); H 1 df with u(·, 0) = u 0 . Furthermore, for all t > 0, (2.20) Proof. Let f (t) = K(t)E(t). Then we can use the product rule and Proposition 2.4 to compute that (2.21) Factoring out a 2νE(t) 2 , we find that Therefore, if f (t) < 6, 912π 4 ν 4 , then f ′ (t) < 0. This implies that if f (0) < 6, 912π 4 ν 4 , then for all 0 < t < T max , we have f (t) < 2, 916π 4 ν 4 . Interpolating between L 2 andḢ 1 , we can see that Sverák, Seregin, and Escauriaza showed in [9] that if T max < +∞, then Therefore, f (0) < 6, 912π 4 ν 4 implies that T max = +∞. Now we will consider the bound on enstrophy globally in time. We know that Fix t > 0. Integrating this differential inequality and making use of the energy inequality, we find Rearranging terms we find that We took t > 0 arbitrary, so this completes the proof.
Similar estimates were considered by Protas and Ayala in [1]. In particular, they proved that if E 0 K 0 < 16π 4 ν 4 27 , then there must be a smooth solution globally in time, and enstrophy is bounded , for all t > 0. By improving the constant for enstrophy growth instantaneously in time, we significantly expand the set of initial data for which we are guaranteed to have global smooth solutions. The initial data must be in H 1 for the product of initial energy and initial enstrophy to be bounded, so the condition in Proposition 2.6 is more restrictive than the condition in the small initial data results forḢ 1 2 [11], L 3 [16], or BM O −1 [17]. However, the product of energy and enstrophy is the most physically relevant of the scale invariant quantities, and so we are able to sharpen the bound on the size initial data for which solutions are guaranteed to be smooth globally in time more effectively in this case by taking advantage of the structure of the nonlinear term. The proofs of the bounds for small initial data inḢ  Tao [30], as would the estimates used by Protas and Ayala. The estimates used to prove Proposition 2.6, on the other hand, take advantage of the structure of the evolution equations for vorticity and strain, and the constraint spaces, and so would not hold with the same constants in Tao's model equation.
We will now prove an immediate corollary of Proposition 2.6, that any solution that blows up in finite time must be bounded away from zero that will be useful later on.
is a mild solution to the Navier-Stokes equation and T max < +∞, then for all 0 ≤ t < T max , (2.28) Proof. We will prove the contrapositive. Suppose that there exists 0 ≤ t < T max such that K(t)E(t) < 6, 912π 4 ν 4 . Then by Proposition 2.6, u(·, t) generates a global smooth solution to the Navier-Stokes equations. Smooth solutions of the Navier-Stokes equations are unique, so if u(·, t) generates a global smooth solution to the Navier-Stokes equations, then so does u 0 .
Using Proposition 2.4, we can also prove an upper bound on blowup time, assuming there is finite time blowup, in terms of the initial energy, and a lower bound on blowup time in terms of the initial enstrophy. We will prove these results below.
Proof. Suppose toward contradiction that 13,824π 4 ν 5 < T max < +∞. We know from the energy equality that We also know from the energy equality that K(t) < K 0 . Combining these two inequalities as well as our hypothesis on T max , we find that Using Proposition 2.6, this implies that if we take u(·, t) to be initial data, it generates a global smooth solution, which contradicts the assumption that T max < +∞. The uniqueness of strong solutions means that if u(·, t) generates a global smooth solution for some 0 < t < T max , then so does u 0 . This contradicts the assumption that T max < +∞, and completes the proof.
Proposition 2.9. For all u 0 ∈Ḣ 1 df , and for all 0 < t < 1,728π 4 ν 3 Proof. Integrating the differential inequality we find that for all 0 < t < 1,728π 4 ν 3 (2.33) Rearranging terms we find that for all 0 < t < 1,728π 4 ν 3 The mild solution can be continued further in time as long as enstrophy is bounded, so this completes the proof.

A logarithmic correction
In order to prove the Theorem 1.9, we will need to prove some bounds on the growth of ω h Ḣ − 1 2 , as well as bound the growth of enstrophy in terms of ω h Ḣ − 1 2 . In order to do this we will need to consider the evolution equation for the horizontal components of vorticity, ω h , which is given in the following proposition.
is a mild solution, and therefore a classical solution, to the Navier-Stokes equation. Then ω h is a classical solution of Proof. Kato and Fujita proved that mild solutions must be smooth [11], so clearly u is a classical solution to the Navier-Stokes equation. Therefore ω = ∇ × u is also smooth and is a classical solution to the vorticity equation: Multiply the vorticity equation through by Next we add and subtract SI h ω. Therefore, Regrouping terms we find that Recall that I h ω = ω h and compute that S h = I h S − SI h , and this completes the proof.
One of the key aspects in our proof is a generalization of the isometry in Proposition 1.3 that tells us S 2 L 2 = 1 2 ω 2 L 2 , to an isometry that involves just one column of S and just two components of ω. In order to state this isometry, we will define the vectors v 1 , v 2 , v 3 as follows.
With these vectors defined, we can restate our identity for enstrophy growth in Proposition 2.1 in terms of v 1 , v 2 , v 3 .
be a mild solution to the Navier-Stokes equation. Then for all 0 ≤ t < T max , we have Proof. We know that v 1 , v 2 , v 3 are the columns of 2S, so by the triple product representation of the determinant of a three by three matrix The three by three determinant is homogeneous of order three, so Recalling from Proposition 2.1 that this completes the proof.
We will now prove an isometry that relates Hilbert norms v 3 and ω h to each other and to ∂ 3 u and ∇u 3 , as well as bounding Hilbert norms of S h by ω h .
Proof. First we observe that (3.14) Next we observe that because ∇ · u = 0, then clearly ∇ · ∂ 3 u = 0. Therefore ∂ 3 u and ∇u 3 are orthogonal inḢ α , so This means we can compute that This completes the first part of the proof. Finally we see that Therefore we can conclude that This completes the proof.
Remark 3.5. Another way to see this isometry, is that In fact, for any fixed vector v ∈ R 3 we will have This is directly related to Proposition 1.3, because This shows that the isometry between the symmetric and anti-symmetric part of the gradient, between strain and vorticity, not only holds overall, but also in any fixed direction.
This isometry, together with the identity for enstrophy growth in Proposition 3.3, will allow us to prove a new bound on the growth of enstrophy in terms of the critical Hilbert norm of ω h . Before we proceed with this estimate, we will note that there is also a generalization of this result in L q . The L q norms of v 3 and ω h are also equivalent, although not necessarily equal.
Proposition 3.6. Fix 1 < q < +∞ and let B q ≥ 1 be the constant from the Helmholtz decomposition, Proposition 1.4. Then for all u ∈Ẇ 1,q Proof. As we have already seen, so clearly Observing that ∂ 3 u = P df (∂ 3 u − ∇u 3 ) , and ∇u 3 = P g (∂ 3 u − ∇u 3 ) , we can apply Proposition 1.4 and find that Recalling that v 3 = ∂ 3 u + ∇u 3 , we apply the triangle inequality and find that We have proven the second inequality. Now we need to show that The argument is essentially the same. Observe that ∂ 3 u = P df v 3 and ∇u 3 = P g v 3 . Therefore from Proposition 1.4, we find that Applying the triangle inequality, we find that This completes the proof.
Proposition 3.7. Taking C 1 and C 2 as in Lemmas 2.3 and 2.5, let Then for all mild solutions to the Navier-Stokes equation Proof. We begin by applying Proposition 3.3, Lemma 3.4, and the duality ofḢ − 1 2 andḢ 1 2 . We find that: Next we apply the fractional Sobolev inequality, the chain rule for gradients, the generalized Hölder inequality, and the Sobolev inequality to find that Finally observe that the vectors v i are the columns of 2S, so Therefore we find that Applying Proposition 1.3 and recalling that 1 This completes the proof of the bound. Now we will prove the second piece. Suppose T max < +∞. Then Therefore, for all ǫ > 0, ω(·, t) L 2 is not nonincreasing on the interval (T max − ǫ, T max ). Therefore, for all ǫ > 0, there exists t ∈ (T max − ǫ, T max ), such that ∂ t ω(·, t) L 2 > 0. Applying the bound we have just proven, this implies that for all ǫ > 0, there exists t ∈ (T max − ǫ, T max ) such that This completes the proof.
We will note that this is theḢ − 1 2 version of a theorem proved in L 3 2 by Chae and Choe in [4]. Their result is the following.
Furthermore, for all 3 2 < q < +∞, let 3 q + 2 p = 2. There exists C q > 0 defending on only q and ν such that for all 0 ≤ t < T max Proposition 3.7 extends the result of Chae and Choe from a lower bound on ω h in L 3 2 near a possible singularity to a lower bound inḢ − 1 2 near a possible singularity. The analysis of the relationship between ω h and v 3 also sheds some light on a relationship between Theorem 3.8 and the following theorem prove by the author in [26].
then there exists D q > 0 depending on only p, q, and ν, such that We will note here that Proposition 3.6 implies that the special case of Theorem 3.9 is equivalent to Chae and Choe's result in Theorem 3.8 for 3 2 < q < +∞, because we have shown that for 1 < q < +∞, ω h L q and ∂ 3 u + ∇u 3 L q are equivalent norms. Theorem 3.9 is more general, however, in that it does not require the strain blow up only in a fixed direction, but also allows the direction to vary.
We previously found a bound for enstrophy growth in terms of ω h Ḣ − 1 2 . The next step will be to prove a bound on the growth of ω h Ḣ − 1 2 using the evolution equation for ω h in Proposition 3.1 and the bounds in Proposition 3.4.
Proof. We begin by using the evolution equation for ω h in Proposition 3.1 to compute that Next we bound the last term using the duality ofḢ 1 andḢ −1 : where we have applied the definition of theḢ 1 to show that (−∆) − 1 2 ω h Ḣ1 = ω h L 2 , and then applied the Sobolev inequality in Lemma 2.5. Applying the triangle inequality, the generalized Hölder inequality, and the fractional Sobolev inequality we can see that Applying Lemma 3.4 we observe that S h Ḣ 1 2 ≤ 1 √ 2 ω h Ḣ 1 2 , and applying Proposition 1.3 we observe that S L 2 = 1 √ 2 ω L 2 . Finally we can conclude that We now turn our attention to the term − (−∆) − 1 2 ω h , (u · ∇)ω h . First we note that u ∈ C ((0, T max ); H ∞ ) , due to the higher regularity of mild solutions, so we have sufficient regularity to integrate by parts. Using the fact that ∇ · u = 0, conclude that Applying the generalized Hölder inequality, the Sobolev inequality, and the isometry in Proposition 1.3, and interpolating betweenḢ −1 andḢ 1 as above, we find that Combining the bounds in (3.71) and (3.77), we find that . Observe that Therefore f has a global max at r 0 = 3M 4ν . This implies that (3.82) Substituting in for M, we find that Multiplying both sides by 2, and substituting in 1 , observe that Applying Grönwall's inequality, this completes the proof.
With this bound, we now have developed all the machinery we need to prove the main result of this paper, Theorem 1.9, which is restated here for the reader's convenience.

85)
u 0 generates a unique, global smooth solution to the Navier-Stokes equation u ∈ C (0, +∞); H 1 df , that is T max = +∞. Note that the smallness condition can be equivalently stated as

86)
and that the constants R 1 and R 2 are taken as in Propositions 3.7 and 3.10.
Proof. We will prove the contrapositive. That is we will show that T max < +∞ implies that Using Proposition 2.6, T max < +∞ implies that K 0 E 0 ≥ 6, 912π 4 ν 4 . This means that Therefore ω(·, t) L 2 cannot be non-decreasing on (0, T max ). There exists 0 <t < T max such that ∂ t ω(·, t) 2 L 2 > 0. By Proposition 3.7, we can conclude that there exists 0 <t < T max such that , so by the intermediate value theorem, there exists It is clear from the intermediate value theorem and the fact that ω 0 Applying Proposition 3.7, this implies that for all t < T, ∂ t ω(·, t) 2 L 2 < 0. Using Proposition 3.10, observe that Using the fact that ω(·, t) L 2 is decreasing on the interval [0, T ], we can pull out a factor of ω 0 2 L 2 , and conclude We know from the energy equality that (3.94) Again using the fact that ω(·, t) L 2 is decreasing on the interval [0, T ], and therefore that ω(·, T ) L 2 < ω 0 L 2 , we may conclude that This means that (3.97) Therefore T max < +∞ implies that This completes the proof.

Boundedness in Besov spaces
Now that we have proved Theorem 3.11, we will consider the size of the set of initial data for which we have proven global regularity in Hilbert and Besov spaces. In this section we will prove Theorem 1.10. To begin with, we will denote the set of initial data satisfying the hypotheses of Theorem 3.11 by Γ 2d .
Definition 4.1. We will define the set Γ 2d ⊂ H 1 df to be the set of almost two dimensional initial data satisfying the hypothesis of Theorem 3.11: Because the proof of Theorem 1.10 will rely heavily on the structure of the vorticity, we will also introduce the set of initial vorticities satisfying the hypothesis of Theorem 1.10. It will be easier to prove our results in terms of vorticity, and then show that the results in terms of velocity are equivalent.
Definition 4.2. We will define the setΓ 2d ⊂ L 2 df ∩Ḣ −1 df to be the set of almost two dimensional initial vorticities satisfying the hypothesis of Theorem 3.11: Note thatΓ 2d is the image of Γ 2d under the curl operator. We will now define Besov spaces with negative indices. .

Lemma 4.5.
Let Ω ǫ ⊂ R 3 be defined by 14) where z = ξ 3 and r = ξ 2 1 + ξ 2 2 are the cylindrical coordinates in Fourier space. Then for all 2 < p ≤ +∞ there exists a constantC p > 0 depending only on p, such that for all f ∈Ḣ Proof. Proceeding as in Proposition 4.4, set 1 p + 1 q = 1 and 1 q = 1 2 + 1 s . Again applying Plancherel's inequality and Hölder's inequality, we find that Note that we can again make the change of variables, ζ = t 1 2 ξ, without changing the domain of integration, just as we did in Proposition 4.4, because Ω ǫ is invariant under multiplication by scalars.
Proof. Fix 2 < p ≤ +∞ and ω ∈Γ 2d . We will split up the domain in Fourier space in two parts, Ω ǫ and Ω c ǫ and consider them separately. Fix 0 < ǫ < 1. Define v and w bŷ It is obvious that ω = v + σ, (4.40) and therefore by the triangle inequality we know that (4.41) We will estimate these two quantities separately. By Theorem 4.5, we find that Applying Proposition 4.4 and Lemma 4.6, we observe that Putting together these bounds on the two pieces we find that Because we took 0 < ǫ < 1 arbitrary, it follows immediately that In order to compute this infinimum we will define Computing the derivative, we find that Therefore, f achieves a global maximum on (0, +∞) at (4.52) We now will consider two cases, when ǫ 0 < 1 and when ǫ 0 ≥ 1. If ǫ 0 < 1, then we have where M p depends only on p. Interpolating betweenḢ −1 and L 2 and using that ω ∈Γ 2d by hypothesis, we find that so we may conclude that Therefore we may conclude that with R p,ν < +∞ depending only on p and ν. Now suppose ǫ 0 ≥ 1. This implies that Applying Theorem 4.4 we conclude that Putting together the case where ǫ 0 < 1 and the case where ǫ 0 ≥ 1, we tin that for all ω ∈Γ 2d , This completes the proof.
In order to show thatΓ 2d is unbounded inḂ Applying the Plancherel Theorem, and the fact that for all ξ ∈ Λ n , |ξ| 2 ≤ 5, we compute that where g is given by Proof. Fix 1 ≤ p ≤ +∞ and v ∈ L p R 3 ; R 3 . We will first define the heat kernel, taking The heat operator e t∆ can be defined in terms of convolution with G as follows: Therefore we can compute that Applying Young's inequality for convolutions we find that where ω = ∇ × u.
Note that we have only definedḂ s p,∞ R 3 , for s < 0. This space is also well defined for 0 ≤ s < 3 2 , but in this case cannot be defined in terms of the heat kernel. This theorem holds for all 2 ≤ p ≤ +∞, but we will only prove the case where 3 < p ≤ +∞. In order to prove the case where 2 ≤ p ≤ 3, we would need to introduce a dyadic decomposition of unity to define the homogeneous Besov space with s > 0, and this would clutter this paper with technical details that are ancillary to the main results. We refer the interested reader to Chapter 2 in [2] for more details.
Proof. Fix p > 3 and u ∈Ḃ −1+ 3 p , ∇·u = 0. We will begin by proving the first bound. Let ω = ∇×u. Then using the properties of the heat semi-group, we can see that Applying Lemma 4.9, we can see that for all t > 0, Therefore we can see from Definition 4.3 that (4.117) We will now prove the second inequality, bounding Besov norms of u in terms of Besov norms of ω. Recall that we can invert ω to obtain u with the formula The inverse Laplacian can be computed using the heat kernel via the following formula: Therefore, applying the Minkowski inequality, Lemma 4.9, and Definition 4.3, we can see that for all t > 0, Therefore, but Definition 4.3, we find that This completes the proof.
As we have already mentioned, the reason the constant goes to infinity here as p → 3, is because the definition of the Besov space we are using breaks down for nonnegative indices s ≥ 0. The equivalence also holds in the range 2 ≤ p ≤ 3, but we would need to introduce a lot of technical details that have little to do with almost two dimensional Navier-Stokes flows in order to define Besov spaces for this range of parameters, so it is left to the reader.
We can now prove Theorem 1.10 from the introduction, which is restated here for the reader's convenience.

Relationship to previous results
In this section we will consider the relationship between the vorticity approach to almost two dimensional initial data developed in section 3 and previous global regularity results for almost two dimensional initial data. Gallagher and Chemin proved in [6] that initial data re-scaled so it varies slowly in one direction must generate global smooth solutions.
be a smooth divergence free vector field on R 3 that belongs, along with all of its derivatives, to L 2 R x 3 ;Ḣ −1 R 2 , and let w 0 be any smooth divergence free vector field. For each ǫ > 0 define the re-scaled initial data by Then there exists ǫ 0 > 0, such that for all 0 < ǫ < ǫ 0 , the initial data u 0,ǫ generates a global smooth solution to the Navier-Stokes equations.
This is often referred to as the well-prepared case, because v 0 3 = 0, and so v 0,ǫ converges to a two dimensional vector field in the sense that for all x ∈ R 3 .
We will also note that global regularity in Theorem 5.1 is not a consequence of Koch and Tataru's theorem on global regularity for small initial data in BM O −1 , because, subject to certain conditions, v 0,ǫ is large in B −1 ∞,∞ , the largest scale-critical space. Gallagher, Chemin, and Paicu generalized this result to the ill-prepared case in [7].
Theorem 5.2. Let u 0 be a divergence free vector field on T 2 × R, and for each ǫ > 0 let our rescaling be given by For all a > 0 there exists ǫ 0 , µ > 0 such that if then for all 0 < ǫ < ǫ 0 , the initial data u 0,ǫ generates a global smooth solution to the Navier-Stokes equation.
This is referred to as the ill-prepared case because whenever u 0 3 is not identically zero, this clearly does not converge to any almost two dimensional vector field. The proof of this result is quite technical, in particular because all control over u 0,ǫ 3 is lost as ǫ → 0. This means that the proofs do not rely on L p or Sobolev space estimates, but are based on controlling regularity via a Banach space, B s that is introduced. The theorem in the paper is actually proved in terms of B 7 2 and the result in terms of H 4 follows as a corollary.
The underlying reason for these technical difficulties is that, in order to maintain the divergence free structure needed for the Navier-Stokes equation, making the solution vary slowly in x 3 requires us to make u 0,ǫ 3 large, so that applying the chain rule, One way to get around this technical difficulty without the restriction that v 0 3 = 0, is to perform the rescaling in terms of the vorticity, rather than the velocity. For a solution to be almost two dimensional, we want both and u 3 to be small and for the solution to vary slowly with respect to x 3 , but the divergence free condition doesn't let us scale both out simultaneously.
On the vorticity side however, a two dimensional flow has its vorticity in the vertical direction, so an almost two dimensional flow corresponds to one in which ω 1 and ω 2 are small, and which varies slowly with respect to x 3 . Take This re-scaling preserves the divergence free condition, because applying the chain rule Furthermore, this is a re-scaling which allows us to to converge to an almost two dimensional initial data without any restrictions such as v 0 3 = 0. Theorem 1.9, is not strong enough to prove there is global regularity for sufficiently small ǫ with this re-scaling, because it is only a logarithmic correction. We will, however prove an analogous result that is slightly weaker in terms of scaling, because it grows more slowly in the critical norms as ǫ → 0, but still becomes large in the critical space L 3 2 ; this result in Theorem 1.12 in the introduction, which is restated here for the reader's convenience.
and define u 0,ǫ using the Biot-Savart law by For all u 0 ∈ H 1 df and for all 0 < a < We note that while Theorem 5.3 is weaker in terms of scaling than Theorem 5.2 proven in [7], it is stronger in the sense that it allows us to take as initial data the re-scalings of arbitrary u 0 ∈ H 1 df , whereas Theorem 5.2 requires that the we re-scale u 0 ∈ H 4 that is also smooth with respect to x 3 . The regularity hypotheses on u 0 in Theorem 5.3 are the weakest available in order to ensure global regularity for initial data rescaled to be almost two dimensional. Unfortunately, however, the rescaled initial data do not become large in the endpoint Besov spaceḂ −1 ∞,∞ , so this is not a genuine large data result, unlike the result proven by Gallagher, Chemin and Paicu [7].
Before proving Theorem 5.3, we will need to state a corollary of Theorem 1.9 that guarantees global regularity purely in terms of L p norms of ω.
u 0 generates a unique, global smooth solution to the Navier-Stokes equation u ∈ C (0, +∞); H 1 df , that is T max = +∞, with C 2 taken as in Lemma 2.5, and R 1 and R 2 taken as in Theorem 1.9.
Proof. This is a corollary of Theorem 1.9. Suppose We know from the fractional Sobolev inequality, Lemma 2.3, that and from the Sobolev inequality, Lemma 2.5, that . (5.14) Therefore we can conclude that This implies that Applying Theorem 1.9, this completes the proof.
Remark 5.5. For all 1 ≤ q < +∞, and for all f ∈ L q R 3 where f ǫ (x) = f (x 1 , x 2 , ǫx 3 ), ǫ > 0. This is an elementary computation for the rescaling of the L q norm in one direction.
We will now prove Theorem 5.3.
Proof. Fix u 0 ∈ H 1 df and 0 < a < . We will prove the result using Corollary 5.4.
Applying Remark 5.5, we find that Similarly we apply Remark 5.5, to compute the other relevant L q norms in Corollary 5.4: ) Using the triangle inequality for norms we can see that Likewise we may compute that Combining these inequalities and factoring out the log (ǫ −a ) 1 4 terms we find that Dividing by R 2 ν 3 and taking the exponential of both sides of this inequality, we find that Combining this with the estimate (5.18), we find that (5.29) We know from the definition of a that a w 0 3 2 Clearly we can see that Therefore, there exists r > 0, such that for all 0 < ǫ < r, because the logarithm grows more slowly than any power. Putting these inequalities together we find that This limit is clearly non-negative, so we can conclude that Therefore there exists ǫ 0 > 0, such that for all 0 < ǫ < ǫ 0 , Applying Corollary 5.4, this means for all 0 < ǫ < ǫ 0 there is a unique global smooth solution for initial data u 0,ǫ ∈ H 1 df . Next we will show that unless ω 0 3 is identically zero, This completes the proof.
Iftimie proved the global existence of smooth solutions for the Navier-Stokes equation with three dimensional initial data that are a perturbation of two dimensional initial data. As we mentioned in the introduction, this is possible on the torus, but not on the whole space, in particular because L 2 T 2 defines a subspace of L 2 T 3 , but L 2 R 2 does not define a subspace of L 2 R 3 because we lose integrability. The precise result Iftime showed is the following [14].
Theorem 5.6. There exists C > 0, such that for all v 0 ∈ L 2 df (T 2 ; R 3 ), and for all w 0 ∈ H there exists a unique, global smooth solution to the Navier-Stokes equation with initial data u 0 = v 0 + w 0 .
In fact, Iftimie proves something slightly stronger. The result still holds if the space H 1 2 is replaced by the anisotrophic space H δ,δ, 1 2 −δ , 0 < δ < 1 2 which is the space given by taking the H 1 2 −δ norm with respect to x 3 , leaving x 1 , x 2 fixed, giving us a function of x 1 and x 2 , then taking the H δ norm with respect to x 2 and so forth. In the range 0 < δ < 1 2 , these spaces strictly contain H 1 2 . This result was also extended to the case of the Navier-Stokes equation with an external force by Gallagher [12], but only where the control in w 0 is in the critical Hilbert spaceḢ 1 2 , not in these more complicated, anisotropic spaces. These anisotropic spaces are quite messy; in particular we will note that for α = 0, H α,α,α = H α T 3 . For this reason, and because the results in this paper deal with Hilbert spaces, we will focus our comparison of Iftimie's result with ours in the setting ofḢ 1 2 . For more details on these anisotropic spaces, see [15]. We will find that Iftimie's result neither implies, nor is implied by, our result, but that they are closely related. In order to compare the results in this thesis to the result proven by Iftimie, it is first necessary to state a version of Theoerem 1.9 on the torus. The result will be essentially the same, although possibly with different constants.
The proof of the this result on the torus is exactly the same as the proof of the result on the whole space. The only reason the constants may be different is because the sharp Sobolev constant may be worse on the torus than the whole space. We will note that when considering solutions to the Navier-Stokes equations on the torus, we include the stipulation that the flow over the whole torus integrates to zero, soû This normalization is necessary in order to mod out constant functions on the torus, so without this stipulation, we would not in fact be able to make use of Sobolev and fractional Sobolev inequalities. In order to relate Theorem 5.6 and Theorem 5.7, we will need to define a projection from three dimensional vector fields to two dimensional vector fields, following the approach of Iftimie [14] and Gallagher [13].
Then for all 1 ≤ q ≤ +∞, P 2d : L q df T 3 → L q df T 2 . In particular, ∇ · P 2d (u) = 0, (5.49) and Proof. Notice that we are projecting onto two dimensional vector fields by taking the average in the vertical direction. First we will observe that P 2d is a bounded linear map from L q to L q . Linearity is clear. As for boundedness, applying Minkowski's inequality, we find , g(x 3 ) = 1, and let 1 p + 1 q = 1, then apply Hölder's inequality to observe Now we need to show that for all u ∈ L q df T 3 , ∇ · P 2d (u) = 0. First we will show this by formal computation for u smooth, and then we will extend by density. Fix u ∈ C ∞ T 3 , ∇·u = 0. Observe that (5.54) Using the fact that ∇·u = 0, we can conclude that ∂ 1 u 1 +∂ 2 u 2 = −∂ 3 u 3 . Applying the fundamental theorem of calculus, and using the fact that u 3 is continuous and periodic, we find We will proceed to proving that ∇·P 2d (u) for all u ∈ L q df T 3 . Note, we here refer to divergence free in the sense of integrating against test functions, as u is not differentiable a priori. Fix As we have shown above ∇ · P 2d (v) = 0, so clearly P 2d (v), ∇f = 0. (5.57) Using the linearity of P 2d observe that Applying Hölder's inequality we find that We know from the bound we have already shown that But ǫ > 0 was arbitrary, so taking ǫ → 0, we find that This completes the proof.
We will also define the projection onto the subspace orthogonal to L 2 df T 2 ; R 3 .
Note that this is well defined, because we have already shown that u ∈ L 2 df T 3 implies that P 2d (u) ∈ L 2 df T 3 , so clearly their difference, u − P 2d (u), is also in this space, which means it is a well defined linear map.
Remark 5.10. Note that Theorem 5.6 can be reformulated in terms of P 2d and P ⊥ 2d as saying there exists C > 0 such that for all u 0 ∈ H 1 2 df T 3 , such that Next we will note that P 2d decomposes the support of the Fourier transform of u into the plane where k 3 = 0 and the rest of Z 3 .
Therefore we see that k 3 = 0 implies thatv(k) = 0. Now we will proceed to the case where k 3 = 0. Observe thatv (k 1 , k 2 , 0) = Recalling the definition of P 2d , we can see that v(k 1 , k 2 , 0) = This Fourier decomposition allows us to control P ⊥ 2d (u) by ∂ 3 u, although in doing so we lose criticality. Note that for all k 3 = 0, k 2 3 ≥ 1, so we can see that Recalling that∂ 3 u(k) = 2πik 3û (k), we can compute that This inequality allows us to prove a corollary of Iftimie's result, Theorem 5.6, that is stated as bound on in terms of the size of ∂ 3 u in H 1 2 , rather than in terms of perturbations of L 2 df T 2 .
Corollary 5.13. There exists C > 0 independent of ν, such that for all u 0 ∈ H Proof. We will take C > 0 as in Theorem 5.6. Suppose u 0 ∈ H nonetheless. Let v 0 = P 2d u 0 and let w 0 = u 0 − P 2d u 0 . From Proposition 5.8, we know that v 0 L 2 (T 2 ) ≤ u 0 L 2 (T 3 ) . (5.78) We also know from Proposition 5.12, that Putting these two inequalities together we find that We should note here that Corollary 5.13 is not equivalent to Iftimie's result Theorem 5.6; the corollary is implied by this result, but does not imply it. That is because Iftimie's result involves controlling P ⊥ 2d u 0 Ḣ 1 2 , which is scale critical, but Corollary 5.13 involves controlling ∂ 3 u Ḣ 1 2 , which is not scale critical.
Corollary 5.13 neither implies, nor is implied by Theorem 5.7, which is the main result of this paper translated to the setting of the torus rather than the whole space. This is because on the torus, as on the whole space, This means that Theorem 5.7 is weaker than Corollary 5.13 in the sense that it requires control on both ∂ 3 u and ∇u 3 , but it is stronger in the sense that it requires control in the critical spaceḢ − 1 2 , rather than the subcritical spaceḢ 1 2 .
In fact we will show that Theorem 5.7 is not implied by Theorem 5.6, because it is not possible to control P ⊥ 2d u 0 , where these are the respective critical Hilbert norms in Theorem 5.6 and Theorem 5.7. This precise result will be as follows.
Proposition 5.14. sup df , in terms of its Fourier transform by u n (k) = a n (n, −1, 0), k = ±(1, n, 1) 0, otherwise , where a n is a normalization factor given by a n = √ n 2 + 2 4π (n 2 + 1) 1 2 . (5.84) It is easy to check that for all n, k ∈ N, k· u n (k) = 0, so ∇·u n = 0, and for each n ∈ N, u n ∈ H 1 2 df T 3 . It is not essential to the proof, but we will also note for the sake of clarity that u n (x) = 2a n (n, −1, 0) cos (2π(x 1 + nx 2 + x 3 )) . (5.85) Note that for all n ∈ N u n 3 = 0, so we have ω n h Ḣ − 1 We know that ∂ 3 u(k) = 2πik 3 u n (k), so we can conclude that We know from Proposition 5.11, that the Fourier transform of P 2d (u) is supported on the plane k 3 = 0 in Z 3 . For all k 1 , k 2 ∈ N, u n (k 1 , k 2 , 0) = 0. This implies that for all n ∈ N, P 2d (u n ) = 0, and therefore P ⊥ 2d (u n ) = u n . Observe that u n 2Ḣ Note that we have shown that for all n ∈ N, ω n h Ḣ − 1 2 = 1, and P ⊥ 2d (u n ) Ḣ 1 2 = √ n 2 + 2. Therefore we may conclude that sup By proving that P 2d (u 0 ) Ḣ 1 2 cannot be controlled by ω n h Ḣ − 1 2 , we have shown definitively that Theorem 5.7 is not a corollary of earlier work by Iftimie and separately by Gallagher, and so this result is new on the torus as well as on the whole space. We will conclude this paper by making a remark that the set of initial data in Theorem 5.7, is unbounded in all scale invariant spaces.
Remark 5.15. Unlike Theorem 1.9 which gives examples of global smooth solutions with arbitrarily large initial data inḢ 1 2 R 3 but not inḂ −1 ∞,∞ R 3 , Theorem 5.7 does give provide examples global smooth solutions with arbitrarily large initial data inḂ −1 ∞,∞ T 3 . This is clear in particular because L 2 T 3 ⊂ L 2 T 3 . If we take u 0 (x) = C(1, −1, 0) cos (2π(x 1 + x 2 )) , (5.96) then ω 0 h = 0, so clearly u 0 satisfies the hypotheses of Theorem 5.7 for all C ∈ R. Taking C large, this gives us an example of large intitial data satisfying the hypotheses of Theorem 5.7. This is not new, of course, because global wellposedness for arbitrarily large initial data is well established in two dimensions. However, by taking small perturbations of fully two dimensional initial data, Theorem 5.7 gives examples of initial data that is large inḂ −1 ∞,∞ T 3 that is not fully two dimensional, but nonetheless generates global smooth solutions. For instance, if we take u 0,n (x) = n(1, −1, 0) cos (2π(x 1 + x 2 )) + exp(−n 5 )(1, −2, 1) cos (2π(x 1 + x 2 + x 3 )) , so there are solutions that are arbitrarily large intitial data that are not fully two dimensional that satisfy the hypothesis of Theorem 5.7. We cannot extend this to the case of the whole space however, because L 2 R 2 ⊂ L 2 R 3 .