GLOBAL REGULARITY FOR A MODEL OF NAVIER-STOKES EQUATIONS WITH LOGARITHMIC SUB-DISSIPATION

. In this paper, we study the global regularity to a three-dimensional logarithmic sub-dissipative Navier-Stokes model. This system takes the form of ∂ t u + ( D − 1 / 2 u ) · ∇ u + ∇ p = −A 2 u , where D and A are Fourier multipliers deﬁned by D = |∇| and A = |∇| ln − 1 / 4 ( e + λ ln( e + |∇| )) with λ ≥ 0. The symbols of the D and A are m ( ξ ) = | ξ | and h ( ξ ) = | ξ | / g ( ξ ) respectively, where g ( ξ ) = ln 1/4 ( e + λ ln( e + | ξ | )), λ ≥ 0. It is clear that for the Navier-Stokes equations, global regularity is true under the assumption that h ( ξ ) = | ξ | α for α ≥ 5 / 4. Here by changing the advection term we greatly weaken the dissipation to h ( ξ ) = | ξ | / g ( ξ ). We prove the global well-posedness for any smooth initial data in H s ( R 3 ), s ≥ 3 by using the energy method.


Introduction. The three-dimensional incompressible Navier-Stokes equations reads in the Eulerian coordinates
where u : [0, T ) × R 3 → R 3 is the velocity field, p : [0, T ) × R 3 → R is the scalar pressure function and u 0 : R 3 → R 3 the given initial data. Here we have normalized the kinematic viscosity coefficient to be 1.
It is well-known that the global regularity problem for the Navier-Stokes equations (1) is a significant open question (see, for instance, [5,21,14,27,15,16,8,2,10,3,13,24,25]). The difficulty in answering this question is that the energy estimate, which is the strongest known coercive a priori estimate, is supercritical with respect to the natural scaling of the Navier-Stokes equations.
The fundamental energy identity of the Navier-Stokes equations reads The natural scaling of the Navier-Stokes equations is defined by The standard scale-transformation methods in [22] have shown that the basic energy equation (2) is not sufficient by itself for an affirmative answer to the global regularity problem of the three-dimensional Navier-Stokes equations. At the same time, the energy identity and the incompressibility condition are very important for preventing negative answers to the problem. In particular, it has been observed that the local structure of the Navier-Stokes equations may lead to a kind of dimension reduction as time approaches the potential singular time [8,10], due to the incompressibility constraint of the fluids. We focus on the most recent work [26]. The Navier-Stokes equations on the Euclidean space R 3 can be expressed in the form where S(u) is an anti-symmetric matrix defined by S(u) := ∇u − (∇u) and R is the Riesz operator defined by R := |∇| −1 ∇. In [26], the author proposed a three-dimensional Navier-Stokes model where S(u) := ∇u − (∇u) , R := |∇| −1 ∇ and A = |∇| ln 1/4 (e + λ ln(e + |∇|)) , λ ≥ 0.
We emphasize that when λ = 0, model (4) is very close to the original Navier-Stokes equation (1) in the sense that the nonlinear term in (1) can be written as R × R × [S(u)u] instead of the one appeared in (4). Moreover, the model obeys the energy identity (2) of the Navier-Stokes equations due to (7) below. Wang [26] proved that model (4) is globally well-posed for any initial data in Sobolev space H s with s ≥ 3. The idea of this paper is in spirit similar to that in [26]. In this work, we take another angle of attack to the global regularity problem. We propose the following model where D = |∇| NAVIER-STOKES MODEL WITH LOGARITHMIC SUB-DISSIPATION   181 and A = |∇| ln 1/4 (e + λ ln(e + |∇|)) , λ ≥ 0.
The symbols of the D and A are m(ξ) = |ξ| and h(ξ) = |ξ|/g(ξ) respectively, where g(ξ) = ln 1/4 (e + λ ln(e + |ξ|)), λ ≥ 0. If λ = 0, then −A 2 is the usual Laplacian operator. If λ > 0, then the model (5) is sub-dissipative. Using the standard energy method [23] and [26], it can be proved that the model (5) is globally well-posed for any initial data in Sobolev space H s with s ≥ 3. The proof of Theorem 2.1 is motivated by Tao's work [23] and Wang's work [26]. Let d ≥ 3. In [23] Tao considered the global Cauchy problem for the generalized Navier-Stokes system for u : is a Fourier multiplier whose symbol h : R d → R + is nonnegative(the case h(ξ) = |ξ| is essentially Navier-Stokes). For the hyper-dissipative Navier-Stokes equations, it is folklore (e.g. [11]) that one has global regularity in the critical and subcritical hyperdissipation regimes h(ξ) = |ξ| α for α ≥ (d + 2)/4. Tao improved this by establishing global regularity under the slightly weaker condition that h(ξ) ≥ |ξ| (d+2)/4 /g(|ξ|) for all sufficiently large ξ and some non-decreasing function g : R + → R + such that These results demonstrate that finer structures of the nonlinearity term in the Navier-Stokes equations are more crucial for the study of this system, in addition to the validity of the energy identity and incompressibility which are the most fundamental properties of Navier-Stokes equations.
The purpose of this paper is to generalize the result for the critical and subcritical hyper-dissipative Navier-Stokes equations to the slightly supercritical and sub-dissipative Navier-Stokes system (the case λ > 0). We emphasize that model (5) is very close to the original Navier-Stokes equations (1). Moreover, the model obeys energy identity (2) of the Navier-Stokes equations due to the energy identity (7) below. Before going any further, we give our motivation to consider this model. Just as the classical 3D Navier-Stokes system, (5) has a scale transform. Indeed, if λ = 0, under the following scaling transformation (5) is invariant. In this sense, as the 2D incompressible Navier-Stoke equations, the L 2 energy is the critical case with respect to the scaling of (5) when λ = 0. This motivates us to expect that model (5) admits a large global solution if the initial data belongs to some Lebesgue or Sobolev spaces without smallness condition.
Before ending this introduction, let us mention that, in [7] Hou and Lei suggested a three-dimensional model of the Navier-Stokes equations which obeys a similar energy law to (2). That model is formulated in terms of a set of new variables related to the angular velocity, the angular vorticity, and the angular stream function. The only difference between the 3D model and the reformulated Navier-Stokes equations in terms of these new variables is that certain convection term in the model is neglected. That 3D model preserves almost all the properties of the full 3D Euler or Navier-Stokes equations. If one could construct singular solutions starting from well-prepared initial data for the model (for instance, see [7] and [9]), then it might yield a more convincing conclusion that finer structures, in particular, the effect of convection terms of the Navier-Stokes equations have to be taken into account for a positive answer of its global regularity problem.
Finally, let's look at some of the previous results for various models for which the energy identity is not available (for example, see [6,18]). In a slightly different direction, finite time blowup was established in [17] for a complexified version of the Navier-Stokes equations for which the energy equality was again unavailable. Further models of the Navier-Stokes type, which obey an energy identity, can be found in [7,12,19,20]. We also mention that for the general 3D incompressible Navier-Stokes equations which possess hyper-dissipation in the horizontal direction, Fang and Han in [4] obtain the global existence result when the initial data belongs to some anisotropic Besov spaces.
The rest of this paper is organized as follows. In section 2, we state the main result of this paper. In section 3, we present the proof of global regularity for model (5).
Notation. For a b, we mean that there is a "harmless" positive constant C which may be different from line to line such that a ≤ Cb.

2.
The main result. The main result of this paper is stated as follows: with div u 0 = 0, then the model system (5) for the incompressible three-dimensional Navier-Stokes equations is globally wellposed.
The theorem demonstrates that, showing finite time singularities for a Navier-Stokes model may not be sufficiently convincing to conjecture that the Navier-Stokes equations itself can develop finite time singularities starting from smooth initial data, even though the model is well-designed so that it satisfies energy identity and incompressibility. Any attempt to negatively resolve the Navier-Stokes global regularity problem in three dimensions also has to consider finer structures on the nonlinear portion of the equations than is provided by the energy identity and incompressibility. At last, it seems an interesting question whether there is a possibility that as time approaches the potential singular time, our model could serve as an approximation of the Navier-Stokes equations for computations and simulations.
Furthermore those results demonstrate that finer structures of the nonlinearity in the Navier-Stokes equations are crucial for the study of this model system, beyond the validity of the energy identity and incompressibility which are the most fundamental properties of the Navier-Stokes equation. Without further understanding of the structure of nonlinearity, any attempt to positively resolve the Navier-Stokes global regularity problem in three dimensions is impossible due to [23] and [26], and any attempt to negatively resolve the same problem for a Navier-Stokes model is also not convincing to yield a negative answer to the global regularity problem of the original Navier-Stokes equations due to the example given in this paper.
3. Proof of Theorem 2.1. Without loss of generality, we only consider the Sobolev space H s for s = 3. Once the theorem is proved for the case of s = 3, it is not difficult to get higher regularity of solutions for any t > 0. The local well-posedness of the Navier-Stokes model (5) with initial data u 0 ∈ H 3 is standard (for example, we may apply the well-known Friedrichs approximation scheme in [1]). As a result of local regularity, we have, for any initial data u 0 ∈ H 3 , there exists a unique solution u to (5) which satisfies We start the proof of Theorem 2.1 by establishing the following a priori estimate. Using the classical energy method, taking the L 2 inner product of the model equation (5) We obtain 1 2 which gives that Taking the L 2 inner product of the model equation (5) 1 with u and integrating over R 3 , we get R 3 (∂ t ∂u · ∂u + ∂(D − 1 2 u · ∇u) · ∂u + ∂∇p · ∂u + A 2 ∂u · ∂u)dx = 0.
That is 1 2 Integrating by parts and using the divergence free property, one has A 2 = 0. Now, we estimate A 1 . We can write Then using Cauchy-Schwarz inequality, we obtain the bound By the arithmetic mean-geometric mean inequality we then have To estimate the second term A −1 (∂D − 1 2 u · ∇u) 2 L 2 , we introduce a parameter N = N (t) ≥ 500, a big number to be determined later. Let us divide A −1 (∂D − 1 2 u· ∇u) 2 L 2 into the high frequency part and the low frequency part.
where P > N and P ≤ N are the Fourier projections to the regions {ξ : |ξ| > N } and {ξ : |ξ| ≤ N } defined by We first deal with the high-frequency part. By Sobolev embedding and Plancherel's Theorem, we obtain that Next, we turn to the low-frequency part. Using the Plancherel's Theorem, Sobolev imbedding and Hölder inequality, we have We further estimate ∇u 2 L 2 . Similarly, we divide ∇u 2 L 2 into the high-frequency part and the low-frequency part. For the high frequency part, one has For the low frequency part, we have So, we obtain that Substituting (14) into (13), we infer that Putting (11), (12), (15) all together, we conclude that Inserting (16) into (10), we have Substituting (17) into (8), we finally arrive at Next, taking the L 2 inner product of the model equation (5) 1 with 2 u and then integrating over R 3 , we have Applying the Leibniz rule to ∂ 2 (D − 1 2 u·∇u), there is one term involving third-order derivatives of u, but the contribution of that term vanishes by integration by parts and the divergence free property. Therefore, we obtain that Then, we estimate A 3 and A 4 . Similar to A 1 , we have and Similarly as in (12), we can estimate the high frequency part by Sobolev embedding and Plancherel's Theorem to obtain that

SHUGUANG SHAO, SHU WANG AND WEN-QING XU
and For the low frequency parts, similarly to (13), we have and Hence, substituting (14) into (24) and (25), we can bound the two estimates by Consequently, by (20), (22) and (26) all together, we finally arrive at Similarly, by (21), (23) and (26) all together, we have that It follows from (19), (27), (28), we finally obtain Finally, taking the L 2 inner product of the model equation (5) 1 with 3 u, similar to (19), we obtain Now, we estimate A 5 , A 6 and A 7 . Similarly as in (10), we have and Similarly, for the high frequency part, we have and For the low frequency parts of A −1 (∂ 3 D − 1 2 u·∇u) 2 L 2 and A −1 (∂D − 1 2 u·∂ 2 ∇u) 2 L 2 , using the Plancherel's Theorem, Sobolev imbedding and Hölder inequality, we have and For the low frequency parts of to estimate that Furthermore, by using (14), we can bound the above three estimates by ln(ln N ) Au 2 L 2 u 2 H 3 + N −4 ln (46) Taking γ to be sufficiently small and N (t) = γ −1 (500 + u 2 H 3 ), using Cauchy-Schwarz inequality, by (46) and (7)  (1 + Au 2 L 2 ) u 2 H 3 ln(500 + u 2 H 3 ) ln ln(500 + u 2 H 3 ) Finally by Gronwall's inequality, we have where the positive function C depends only on t and u 0 H 3 which is finite for any 0 ≤ t < ∞.
We complete the proof of Theorem 2.1.