Regularity of 3D axisymmetric Navier-Stokes equations

In this paper, we study the three-dimensional axisymmetric Navier-Stokes system with nonzero swirl. By establishing a new key inequality for the pair $(\frac{\omega^{r}}{r},\frac{\omega^{\theta}}{r})$, we get several Prodi-Serrin type regularity criteria based on the angular velocity, $u^\theta$. Moreover, we obtain the global well-posedness result if the initial angular velocity $u_{0}^{\theta}$ is appropriate small in the critical space $L^{3}(\R^{3})$. Furthermore, we also get several Prodi-Serrin type regularity criteria based on one component of the solutions, say $\omega^3$ or $u^3$.


Introduction
Consider the initial value problem of 3D Navier-Stokes equations: where u(t, x) = (u 1 , u 2 , u 3 ), p(t, x) and u 0 denote, respectively, the fluid velocity field, the pressure, and the given initial velocity field.
For given u 0 ∈ L 2 (R 3 ) with div u 0 = 0 in the sense of distribution, a global weak solution u to the Navier-Stokes equations was constructed by Leray [27] and Hopf [19], which is called Leray-Hopf weak solution. Regularity of such Leray-Hopf weak solution in three dimension is one of the most outstanding open problems in the mathematical fluid mechanics.
Researchers are interested in the classical problem of finding sufficient conditions for the weak solutions such that they become regular. The important result is usually referred as Prodi-Serrin (P-S) conditions (see [11,12,17,35,36,37,38]), i.e. if additional the weak solution u belongs to L p,q T , where 2 p + 3 q ≤ 1, 3 ≤ q ≤ ∞, then the weak solution becomes regular. Hugo Beirão da Veiga [3] proved a P-S type result with two components of u. For the one component case, Y. Zhou and Pokorný [41] obtained the regularity criterion by imposing the integrability of single component u 3 of the velocity field. Furthermore, Cao and Titi [5] established the regularity criterion involving only one entry of the velocity gradient tensor, likely ∂ 3 u 3 . However, the integral condition here is not optimal in the sense of scaling considerations. For such considerations, people also work on the regularity criteria involving one component of u and one other component, say velocity gradient tensor, likely [34]. Recently, the second author and Chenyin Qian [13,14,15] developed the arguments above. They [15] got several almost critical regularity conditions such that the weak solutions of the 3D Navier-Stokes equations become regular, based on one component of the solutions, say u 3 and ∂ 3 u 3 .
Considering the axisymmetric Navier-Stokes equations, there is a scaling invariant quantity ru θ , see [33,29] etc. So, it is very important to consider the critical regularity conditions for u θ for the axisymmetric Navier-Stokes equations. Here, we assume that a solution u of the system (1.1) of the form u(t, x) = u r (t, r, x 3 )e r + u θ (t, r, x 3 )e θ + u 3 (t, r, x 3 )e 3 , where e r = ( In the above, u θ is called the angular velocity. For the axisymmetric solutions of Navier-stokes system, we can equivalently reformulate (1.1) as where we denote the convection derivativeD Dt as For the axisymmetric velocity field u, we can also compute the vorticity ω = curl u as follows, Let b = u r e r + u 3 e 3 . Then, we have that div b = 0 and curl b = ω θ e θ .
When the angular velocity u θ is not trivial, the global well-posedness problem is still open. Recently, tremendous efforts and interesting progress have been made on the regularity problem of the axisymmetric Navier-Stokes equations [6,7,8,9,20,22,23,25,33,40]. P. Zhang and the third author [40] investigated the global well-posedness with various types of smallness conditions on the initial angular velocity u θ 0 of the initial velocity field. In [7,8], Chen, Strain, Tsai and Yau proved that the suitable weak solutions are smooth if the velocity field u satisfies r|u| ≤ C < ∞. Applying the Liouville type theorem for the ancient solutions of Navier-Stokes equations, Zhen Lei and Qi S. Zhang [25] obtained the similar result b ∈ L ∞ T (BMO −1 ). Moreover, there are many significant results under the sufficient condition for regularity of axially symmetric solution of type ω θ ∈ L p,q T , 2 p + 3 q ≤ 2, 3 2 < q < ∞ in [6], and ω θ ∈ L 1 ((0, T );Ḃ 0 ∞,∞ ) in [9]. And concerning on one velocity component, it has been shown that the axisymmetric solution is smooth in (0, [23]. In particular, the authors in [23] also considered the conditions on u θ , The following regularity criteria of u θ greatly develop the corresponding regularity criteria in [22,23,33,40]. Theorem 1.1. Let u be an axisymmetric weak solution of the Navier-Stokes equations (1.1) with the axisymmetric initial data u 0 ∈ H 2 (R 3 ) and div u 0 = 0. If one of following conditions holds true , and there exist α > 0 and sufficient small ε > 0 such that To prove it, the key point is that we find the pair (Φ, Γ) = ( ω r r , ω θ r ) satisfying the following equations (1.6) and the following identities: In fact, Thomas Y. Hou and Congming Li [21] introduced a 1-dimensional model that firstorder approximates the Navier-Stokes equations, which just corresponds to the zero right hand side of above identities. In additional, the following two technical points are also important: the first is that we give the explicit expression of u r r by Γ, see Lemma 2.3; the second is that we establish a general Sobolev-Hardy inequality, see Lemma 2.4. By above techniques, one reduces the problems to estimating certain terms of particular forms. For example, we need to control the term . From Hölder's inequality, we have Using the general Sobolev-Hardy inequality in Lemma 2.4, we can bound Combining the explicit expression of u r r by Γ in Lemma 2.3, one can bound The key point is that one cannot bounded f r 2 by f H 1 . In Theorem 1.1, we add a small assumption in the case The reason is that we cannot using Gronwall's inequality in the case To overcome this small assumption, one possible method should be the De Giorgi type argument or Nash type method (like [25]). But, using the De Giorgi type argument or Nash type method, one needs some additional condition on b to control the convection terms. In our proof, we just use the energy method and divergence free condition to estimate the system (1.6), and the convection terms are not trouble (see (3.2)). Remark 1.2. Theorem 1.1 tells us that if ru θ is Hölder continuity at the variable r uniformly, i.e. there exist a α > 0 and constant C, such that then u is regular. And the authors in [7,8,25] drawn a similar argument by using the Nash-Moser method.
then the solution u is smooth in (0, T ] × R 3 . In [40], P. Zhang and the third author obtained one special regularity criterion u θ ∈ L 4,6 T . From Theorem 1.1 and the interpolation theorem, we have that if ru θ 0 ∈ L ∞ and the angular velocity u θ satisfies Moreover, we will prove a global regularity theorem only on that the initial angular velocity u θ 0 is appropriately small in the critical space L 3 (R 3 ).
Theorem 1.2. Assume that the axisymmetric initial data u 0 ∈ H 2 (R 3 ) and div u 0 = 0. There exist positive constants C 0 , C 1 and C 2 , such that if the initial angular velocity u θ 0 satisfies We also respective investigate the regularity criterion of the vorticity component ω 3 or the velocity component u 3 by introducing the potential ψ defined in [29]. Theorem 1.3. Let u be an axisymmetric weak solution of the Navier-Stokes equations (1.1) with the axisymmetric initial data u 0 ∈ H 2 (R 3 ) and div u 0 = 0. If ru θ 0 ∈ L ∞ (R 3 ) and ω 3 ∈ L p,q T , Theorem 1.4. Let u be an axisymmetric weak solution of the Navier-Stokes equations (1.1) with the axisymmetric initial data u 0 ∈ H 2 (R 3 ) and div u 0 = 0. If u 3 satisfies one of following conditions, , and there exist β > 0 and sufficient small ε 1 > 0 such that Since ψ is bounded if r|u 3 | ≤ C, we deduce following corollary from [25].
Corollary 1.5. Let u be an axisymmetric suitable weak solution of the Navier-Stokes equations (1.1) with the axisymmetric initial data u 0 ∈ L 2 (R 3 ), div u 0 = 0, and ru θ In Section 2, we establish some important lemmas for the use of proof, like the explicit expression of u r r by ω θ r in Lemma 2.3, a general Sobolev-Hardy inequality in Lemma 2.4. Then we will prove the regularity criteria by u θ , ω 3 and u 3 , respectively, in Section 3. At the end, we prove the global well-posedness result in Theorem 1.2 in Section 4.
Notations. We introduce the Banach space L p,q T , equipped with norm
We will give some useful estimates in the axisymmetric Navier-Stokes equations, and refer to [33,29] for details. Note∇ = (∂ r , ∂ 3 ), and Lemma 2.2. Assume u is the smooth axisymmetric solution of (1.1) on [0, T ], with the initial data u 0 , and curl u = ω, then

1)
iii) there exists a constant C = C(q), such that for 1 < q < ∞, Furthermore, we extend the following argument, which is widely used in various equations with free swirl (See [1,32]), to the axisymmetric solutions with nonzero swirl. Lemma 2.3. Assume u is the smooth axisymmetric solution of (1.1), and curl u = ω, then then we get In addition, using the polar coordinates x 1 = r cos θ, x 2 = r sin θ, we obtain Using the L q -boundedness of Riesz operator, we easily obtain (2.4)-(2.5).
Then, we give a general Sobolev-Hardy inequality. Badiale and Tarantello proved the case q * = q(N −s) N −q in [2] (Theorem 2.1). For the convenience of reader's reading, we give the proof by another method.
) be the unique axisymmetric solution of the Navier-Stokes equations with the axisymmetric initial data u 0 ∈ H 2 (R 3 ) and div u 0 = 0. If in addition, T < ∞ and u θ r L 4,4 T < ∞, then u can be continued beyond T .

Proof of the regularity criteria
It is well-known that: if the axisymmetric initial data u 0 ∈ H 2 (R 3 ) and div u 0 = 0, we can construct a global axisymmetric weak solution u (see [4,6]), satisfying the energy inequality, Moreover, the system (1.1) has a local unique solution u on [0, T * ), satisfying u ∈ C([0, T * ); where T * is the earliest blow-up point (see [10] for instance).
) be the unique axisymmetric solution of the Navier-Stokes equations with the axisymmetric initial data u 0 ∈ H 2 (R 3 ) and div u 0 = 0. If in addition, T < ∞ and sup t∈[0,T ) Γ L ∞,2 t < ∞, then u can be continued beyond T .
Assume that T * ≤ T . From (1.7) and (1.8), integrating them over R 3 respectively, we obtain for all t ∈ [0, T * ) where From (3.2) and (3.3), we get for all t ∈ [0, T * ) Then, we estimate I 1 , I 2 , I 3 in the following two cases respectively.
Using the fact that ru θ ∈ L ∞,∞ T and the interpolation theorem, we obtain that u θ ∈ L 2p,2q T . Then u is regular, by Theorem 1.1.

Proof of Theorem 1.4
Assume that T * ≤ T . Applying Lemma 2.2, the potential function ψ satisfies u r = −∂ 3 ψ, ru 3 = ∂ r (rψ). (3.16) From (3.16), we get ψ (3.17) Applying Minkowski's inequality, we obtain Multiplying the u θ equation of (1.2) 2 by (u θ ) 3 , and integrating the resulting equation over R 3 , using the integration by parts, Hölder's inequality and the Sobolev embedding theorem, we have for all t ∈ [0, T * ). Applying Gronwall's inequality, we obtain For a small constant ε 1 > 0 given in (3.23), there exists β > 0 such that (3.21) Using the similar arguments as that in the proof of (3.19), we have for all t ∈ [0, T * ) Since Using the similar argument as that in the proof of Theorem 1.1, we have that u can be continued beyond T * , which contradicts with the definition of T * . Thus, T * > T , which finishes Theorem 1.4.
Proof of Corollary 1.5. From Theorem 1.4 in [25], we get that u is regular in (0, T ] × R 3 .