GLOBAL EXISTENCE OF STRONG SOLUTION TO 3D PERIODIC NAVIER-STOKES EQUATIONS

The purpose of this paper is to bring to light a method through which the global in time existence for arbitrary large in H1 initial data of a strong solution to 3D periodic Navier-Stokes equations follows. The method consists of subdividing the time interval of existence into smaller sub-intervals carefully chosen. These sub-intervals are chosen based on the hypothesis that for any wavenumber m, one can find an interval of time on which the energy quantized in low-frequency components (up to m) of the solution u is lesser than the energy quantized in high-frequency components (down to m) or otherwise the opposite. We associate then a suitable number m to each one of the intervals and we prove that the norm ‖u(t)‖H1 is bounded in both mentioned cases. The process can be continued until reaching the maximal time of existence Tmax which yields the global in time existence of strong solution.


Introduction
Let us consider the following incompressible Navier-Stokes equations: where the constant ν > 0 is the viscosity of the fluid, and T 3 = R 3 /Z 3 is the threedimensional torus with periodic boundary conditions. Here u is a three-dimensional vector field u = (u 1 , u 2 , u 3 ) representing the velocity of the fluid, and p is a scalar denoting the pressure, both are unknown functions of the time variable t and space variable x. We recall that the pressure can be eliminated by projecting (N SE) onto the space of free divergence vector fields, using the Leray projector Thus, it will be convenient using the following equivalent system We define the Sobolev spaces H s (T 3 ) for s ≥ 0 by the Fourier expansion (1 + |k| 2s )|û(k, t)| 2 andû (k, t) = T 3 u(x)e −ikx dx.
We will also use the following function spaces:

it was proven by Leray and Hopf that there exists a global weak solution
) of the Navier-Stokes equations satisfying the initial condition u 0 . In particular, u satisfies the energy inequality This result was proved by Hopf [6] as a generalisation of a previous existence theorem due to Leray [8] for the whole space R 3 . It is also known that local in time strong solutions exist on the whole space due to Leray [8], while the case of a bounded domain is due to Kiselev and Ladyzhenskaya [7]. Theorem 1.2. There is a constant C > 0 such that any initial condition u 0 ∈ H 1 σ (T 3 ) gives rise to a strong solution of the Navier-Stokes equations The existence of global in time strong solution is known to occur for small initial data due to Fujita and Kato [5] and Chemin [2]. However, it remains the major open problem as to whether these solutions can be extended to be global in time for arbitrary large in H 1 initial data. Originally, the problem is the question of global existence of smooth solutions to the Navier-Stokes equations satisfying bounded energy condition (i.e.: u ∈ C ∞ (R + × T 3 ) and T 3 |u(x)| 2 dx < ∞) or otherwise a breakdown. The official description has been given by Fefferman in [4]. The official Clay Millennium problem is to give a proof of one of the four following statements: (A) Existence and smoothness of Navier-Stokes solutions on R 3 (B) Existence and smoothness of Navier-Stokes solutions on R 3 /Z 3 (C) Breakdown of Navier-Stokes solutions on R 3 (C) Breakdown of Navier-Stokes solutions on R 3 /Z 3 In this paper, we prove the statement (B) which can be alternatively formulated as follows: The method used to extend the solution into a global one is to prove that on an interval of strictly positive length [t 0 , t 1 ] ⊂ (0, T max ) under a certain condition on k∈Z 3 |û(k, t)| among two possible ones: , the solution will be controlled in H 1 by a function defined in terms of time t, ∇u(t 0 ) L 2 , u 0 L 2 and a finite number m (depending on the viscosity ν and . We continue then in this vein until reaching T max . To be more precise, we subdivide the interval (0, T max ) into a series of successive sub-intervals each of them is akin to [t 0 , t 1 ], i.e.: on each of them either condition 1 or 2 holds. We quote the following two results, the proof of which is given in [9] and based on that of (Theorem 10.6 [3]).

Theorem 1.4. Let u be a strong solution of the Navier-Stokes equations
The Theorem 1.4 together with the following lemma constitute a cornerstone in establishing the proof of Theorem 1.3.
The rest of the paper is dedicated to give the proof of Theorem 1.3.

The proof
The analysis can be started by sketching the procedure through which the existence of local in time strong solution to (N S) follows. To this end, let P n be the projection onto the Fourier modes of order up to n ∈ N, that is Let u n = P n u be the solution to For some T n , there exists a solution u n ∈ C ∞ ([0, T n )×T 3 ) to this finite-dimensional locally-Lipschitz system of ODEs. We take the L 2 -inner product of the first equation in (N S n ) against −∆u n to obtain where we used Hölder's inequality together with Agmon's inequality [1] and the Poincaré inequality. Using Young's inequality with exponents 4 and 4/3 yields where c is a positive constant that does not depend on n. It turns out that By comparing the function ∇u n (t) 2 L 2 with the solution of the ODE: we infer that as long as 0 ≤ t < 1 2c ∇u0 4 L 2 , the following holds . (2.1) Estimate (2.1) tells us that an initial data as large as ∇u 0 L 2 gives rise to a solution u that would remain bounded on the interval [0, However, what happens after time T max is unknown. We turn now to the question of whether a local in time strong solution can be extended into a global solution. To this end, let us make estimates directly for u instead of using the Galerkin approximation. We know that u 0 gives rise to a strong solution that exists at least on a certain time interval [0, T max ). On this time interval for each time t ∈ (0, T max ) we take the L 2 -inner product of (N S) against −∆u, we obtain The Fourier expansion of u(x, t) is given by For a certain number m (to be discussed later on), we have Two possible natural cases may occur. The first is when the major amount of energy at the instant t is quantized in low-frequency components. This case can be represented by The second case is when the major amount of energy at time t is quantized in high-frequency components. That is to say We state here the Agmon's inequality [1] which reads: By Theorem 1.4 we have u ∈ C([0, T max ); H 1 ) and C((0, T max ); H 2 ), then by a continuity argument one can always find at least a small interval of strictly positive length [t 0 , t 1 ] ⊂ (0, T max ) on which either (2.2) or (2.3) occurs for any positive number m. Since [t 0 , t 1 ] will serve as a test interval to examine case (2.2) and case (2.3), one can choose without loss of generality t 0 very close to the instant zero. To be more precise, let t 0 be the instant of time immediately after t = 0, there exists t 1 > t 0 such that we have either (2.2) for all t ∈ [t 0 , t 1 ] or (2.3) for all t ∈ [t 0 , t 1 ]. We need also to make use of Lemma 1.5, which combined with the fact that Property (2.4) is useful because it assures the smoothness of k∈Z 3 |û(k, t)| with respect to time and hence that of |k|≤m |û(k, t)| and |k|>m |û(k, t)|. This prevents the abrupt bends of the function t → F m (t) = |k|≤m |û(k, t)| − |k|>m |û(k, t)|. Let us now discuss both cases on [t 0 , t 1 ]. To be more precise, if (2.2) holds true on [t 0 , t 1 ] what will happen and if (2.3) holds true on [t 0 , t 1 ] what will happen. Case 1: By using (2.2), the Cauchy-Schwarz inequality and Young's inequality, we get where C(m) = 2 |k|≤m 1 ν . By using the energy inequality for weak solutions (1.1) and dropping the viscous term from both sides above, we obtain The Gronwall's inequality yields

Case 2:
By using (2.3) and the Cauchy-Schwarz inequality we infer that Since |k|>m |k| 4 |û(k, t)| 2 ≤ k∈Z 3 |k| 4 |û(k, t)| 2 = ∆u(t) 2 L 2 (T 3 ) , it turns out In such a way, the factor {ν − 2c(m) ∇u(t) L 2 (T 3 ) } would still positive at least over a short interval of time [t 0 , τ 1 ] ⊂ [t 0 , t 1 ]. Consequently, it turns out that But as ∇u(t) L 2 (T 3 ) is continuous on [t 0 , t 1 ], we obtain Thus, the condition on m has been determined successfully. To summarize, we have proved that there exists t 1 > t 0 such that we have either Continuing in this vein, in the next interval we know already that m must be as large as c(m) < ν 2 ∇u(t 1 ) −1 L 2 which guarantees by continuity that in case (2.3) the function ∇u(t) L 2 is non-increasing on this interval. There exists then t 2 > t 1 such that Repeating this process as many times as needed to obtain [t 0 , T max −ε] = ∪ N j=0 [t j , t j+1 ] (where ε > 0 is an arbitrary small constant and [t j , t j+1 ] are successive intervals), such that for all t ∈ [t j , t j+1 ] we have either We remark that the number m may change from an interval [t j , t j+1 ] to the next one [t j+1 , t j+2 ], but it remains finite as long as ∇u(t j+1 ) L 2 is finite which is guaranteed by the estimates (2.6) or (2.7). This process would certainly control the norm ∇u(t) 2 L 2 and rules out the blowup of u in H 1 (T 3 ) as t approaches T max . Therefore, the solution u can be extended into a global in time strong solution.

Discussion
At this point, one may ask the question under which condition the norm ∇u(t) L 2 keeps decreasing for all positive time t ∈ R + . In fact, this is possible when the distribution of energy in the initial data is extremely unbalanced (i.e. |k|≤m |û(k, 0)| << |k|>m |û(k, 0)|). In that case, by smoothness of the function k∈Z |û(k, t)| with respect to time, condition (2.3) keeps for a long interval of time until potentially ∇u(t) L 2 satisfies the smallness condition of [5]. Another aspect to discuss here is the motivation behind choosing the instant t 0 very close to zero. In fact, by doing so one can ensure via (2.1) the closeness of ∇u(t 0 ) L 2 to ∇u 0 L 2 while holding the necessary regularity (u(t 0 ) ∈ H p (T 3 ) for all p ∈ N). This also guarantees the minimum worsening to ∇u(t 1 ) L 2 in case (2.2). However, it is needless to say that this was optional and that one can choose any instant t 0 ∈ [0, T max ) as initial time.

Conclusion
We have already proved that the local in time strong solution to (N S) can be extended to be global in time strong solution. This was done via making estimates to u in H 1 on a series of time intervals requiring that the function t → F m (t) = |k|≤m |û(k, t)| − |k|>m |û(k, t)| keeps its sign constant (either positive or negative) on each of them. It is worth noting that when the major amount of energy is located in high-frequency components (i.e. F m (t) ≤ 0), the norm ∇u(t) L 2 decreases with time. This is in fact consistent with the phenomenology of the turbulent cascade which states that energy is dissipated at the small scales (i.e. higher frequencies).