On some regularity criteria for axisymmetric Navier-Stokes equations

We point out some criteria that imply regularity of axisymmetric solutions to Navier-Stokes equations. We show that boundedness of $\|{v_{r}}/{\sqrt{r^3}}\|_{L_2({\rm R}^3\times (0,T))}$ as well as boundedness of $\|{\omega_{\varphi}}/{\sqrt{r}}\|_{L_2({\rm R}^3\times (0,T))},$ where $v_r$ is the radial component of velocity and $\omega_{\varphi}$ is the angular component of vorticity, imply regularity of weak solutions.


Introduction
We consider the Cauchy problem to the three-dimensional axisymmetric Navier-Stokes equations: (1.1) where x = (x 1 , x 2 , x 3 ), v is the velocity of the fluid motion with v(x, t) = (v 1 (x, t), v 2 (x, t), v 3 (x, t)) ∈ R 3 , p = p(x, t) ∈ R 1 denotes the pressure, and v 0 is given initial velocity field.
The first papers concerning regularity of axially symmetric solutions to the Navier-Stokes equations were independently proved by Ladyzhenskaya [L] and Yudovich-Ukhovskij [YU] in 1968. In these papers axisymmetric solutions without swirl were considered. In the period 1999-2002 arised many papers concerning sufficient conditions on regularity of axisymmetric solutions ( [CL], [NP1], [NP2], [LMNP]).
Especially, conditions on one coordinate of velocity were considered. Recently there are many papers dealing with new sufficient conditions (see references of Lei and Zhang [LZ]).
Our aim is to derive some criteria guaranteeing regularity of solutions to the axisymmetric Navier-Stokes equations. By the regular solutions we mean smooth weak solutions obtained by the standard increasing regularity technique for smooth initial data.There is a lot of criteria for regularity of axisymmetric solutions (see [CFZ], [NP1], [KP], [KPZ], [CT], [LZ], [Z] and the literature cited in these papers). In Section 2 we recall only such criteria that are useful for our analysis.
Then the cylindrical components of velocity and vorticity (ω = rot v) for axisymmetric solutions (therefore, solutions independent of ϕ) are represented as v = v r (r, z, t)ē r + v ϕ (r, z, t)ē ϕ + v z (r, z, t)ē z and ω = ω r (r, z, t)ē r + ω ϕ (r, z, t)ē ϕ + ω z (r, z, t)ē z where v r , v ϕ , v z are radial, angular and axial components of velocity. The axisymmetric motion can be described by the three quantities: v ϕ , ω ϕ and the stream potential ψ which are solutions to the following equations: It is very convenient to introduce quantities u 1 , ω 1 , ψ 1 by the relations v ϕ = ru 1 , ω ϕ = rω 1 , ψ = rψ 1 Then equations (1.2) simplify to We prove the following regularity criteria (see (1.7), (1.8)) which are scaling invariant: Theorem 1. 1. Let (v, p) be an axisymmetric solution to the Navier-Stokes equations (1.1) with the axisymmetric initial data and div v(0) = 0.

Assume that
3. Assume that there exists constant c 1 such that [CKN] there is no singular points. InR 3 r 0 = {x ∈ R 3 , r > r 0 } the axisymmetric problem (1.1) is two-dimensional so local regularity of v is evident.
Theorem 2. Let the assumptions 1,2 of Theorem 1 hold. If Remark 1.2. For r < ∞, the assumption is fulfilled by Lemma 2.6 and Lemma 2.8 because for r ≤ c 0 holds

Notation and auxiliary results
By L p (R N ), p ∈ [1, ∞], we denote the Lebesgue space of integrable functions. By L p,q (R 3 × (0, T )) we denote the anisotropic Lebesgue space with the following finite norm where [l] is the integer part of l.
By H s (R 3 ), s ∈ N 0 = N ∪ {0} we denote the Sobolev space W s 2 (R 3 ). Lemma 2.1. There exists a weak solution to problem (1.1) such that v ∈ L ∞ (0, T ; L 2 (R 3 )) ∩ L 2 (0, T ; H 1 (R 3 )) and the following estimate holds In the case of axisymmetric solutions the energy inequality (2.1) takes the form (2.2) Proof. Equations (1.1) 1,2 for the axially symmetric solutions assume the form where v · ∇ = v r ∂ r + v z ∂ z , ∆u = 1 r (ru ,r ) ,r + u ,zz . Let I = {(ϕ, r, z) : r = 0} denote the axis of symmetry. Define the space X as the closure of C ∞ 0 (R 3 \ I) in the X norm. Then, we are looking for a priori estimate for functions v ∈ X.
Multiplying (2.4) by v ϕ and integrating over R 3 yields Multiplying (2.5) by v z and integrating over R 3 we obtain 1 2 Adding the above equations and using (2.6) we obtain 1 2 Integrating (2.7) with respect to time from 0 to t, t ≤ T, yields . This ends the proof.
To derive energy estimates in the proof of Lemma 2.1 we use the ideas from the proof of Theorem 3.1 from [T], Ch.3. The notion of a suitable weak solution was introduced by L. Caffarelli, R. Kohn and L. Nirenberg in the famous paper [CKN]. Our aim is to show that either (1.7) or (1.8) implies that a suitable weak solution to problem (1.1) does not contain singular points.This means that (v, p) is a regular solution to (1.1). In other words it means that if Therefore v has no singular points (see [CKN]). To show this we use results of J. Neustupa, M. Pokorny and O. Kreml (see [NP1], [NP2], [KP]). To clarify presentation we recall the results.
From [NP2] it follows that Then v has no singular points in D.
Simplifying we have .
By the Gronwall lemma we have .
Passing with ε → 0 we derive (2.10) and conclude the proof.
Introduce the quantities (Φ, Γ) = ω r r , ω ϕ r which are solutions to the equations Remark 2.9. In the proof of Theorem 1.1, Case 1 in [CFZ] there is derived the formula (3.8) in [CFZ] in the form Let us recall some properties of weak solutions to (1.1).

Sufficient conditions for regularity
Let u α = u r α , α ∈ (0, 1). Then u α satisfies Then (3.2) Proof. Multiplying (3.1) by u α |u α | s−2 , integrating the result over R 3 and using that u α ∈ C ∞ 0 (R 3 \ I), we obtain 1 s The second term in the l.h.s. of the above equality can be estimated by Using Lemma 2.4, the second integral in I is bounded by Employing this estimate, with ε = να(2 − α), integrating the result with respect to time and using the density argument, yields Let us assume that the r.hs. of (3.3) is bounded by a constant c 2 . Then Lemma 2.10 implies Comparing the above approach with (2.15) we have that d = 1 − α and the regularity criterion has the form Therefore αs = 3.
In view of (2.16) we have Consider Lemma 2.3. Let s = 4. Then r = 8, so 3 4 + 2 8 = 1 Since 3 4 < 1, v ϕ satisfies assumptions of Lemma 2.3. Hence v has no singular points in R 3 r 0 × (0, T ). Next we show that axisymmetric solutions to problem (1.1) do not have singular points in the region located in a positive distance from the axis of symmetry.
In [CKN] is shown that singular points of v in the axisymmetric case may appear on the axis of symmetry only. Therefore in any region located in a positive distance from the axis of symmetry there is no singular points of v. However, we want to show that statement explicitly. Therefore, we proceed as follows.
Finally we estimate the last two integrals on the r.h.s. of (3.7). We write them in the form ,t) and the first factor is bounded in view of Remark 2.5 and the second factor is absorbed by (3.8) for 2(s − 2) ≤ 5 2 s which holds for s ≤ 12. Finally where the first integral is absorbed by the second term on the l.h.s. of (3.7) and the second is bounded by The first factor in I 3 is bounded in virtue of Remark 2.5 and the second factor is absorbed by (3.8) if 2(s−2) ≤ 5 3 s so s ≤ 12. Summarizing, we obtain the estimate v ϕ L 20 3 (R 3 ×(0,t)) ≤ c (3.10) We observe that the estimate (3.10) above is not strong enough to apply Lemma 2.2(1) because for s = r it is required that s ≥ 50 7 . To increase regularity we introduce a new smooth cut-off function where we can use (3.10). Hence increasing regularity of (3.10) can be achieved to meet assumptions of Lemma 2.2(1). This concludes the proof.
Then there exists a constant c such that To simplify further considerations we introduce the notation u = ψ ,z , f = ω ϕ Then (3.12) takes the form Recall that I is the axis of symmetry. Let C ∞ 0 (R 3 \ I) be the set of smooth functions vanishing near I and outside a compact set.
Recall that functions from H 1 0 (R 3 ) vanish on I. By the weak solution to (3.13) we mean a function u ∈ H 1 0 (R 3 ) satisfying the integral identity which holds for any smooth function χ ∈ C ∞ 0 (R 3 \ I) belonging to H 1 0 (R 3 ). Introducing the scalar product we can write (3.14) in the following short form uvdx.

Hence by the Riesz Theorem there exists
. Therefore there exists a solution to the integral identity (3.15) such that u = F ∈ H 1 0 (R 3 ) and the estimate holds . It is clear that the solution is unique.
Since u vanishes for r = 0, u = ψ ,z so ψ| r=0 = 0 also. Therefore we can look for approximate weak solution satisfying the integral identity where R 3 ε = {x ∈ R 3 : r > ε}, ε > 0. Let χ = ψ,z r α . Then recalling notation u = ψ ,z , f = ω ϕ identity (3.16) takes the form ( 3.17) The second integral on the l.h.s. of (3.17) takes the form Using this in (3.17) and applying the Hölder and Young inequalities to the r.h.s. term yields We have to emphasize that ψ in (3.18) is an approximate function. This should be denoted with ψ ε but we omitted it for simplicity.
Passing with ε → 0, setting α = 1 and integrating this inequality with respect to time implies (3.11). This concludes the proof.
To make statement more explicit we obtain from (3.19) for r < r 0 and from Step 5 of the proof of Theorem 1 from [NP2] that ω L∞(0,T ;L 2 (R 3 r 0 )) ≤ C(data) where data are data from the assumptions of the Step 1 of the proof of Theorem 1 from [NP2].
Proof of Theorem 2 follows from Theorem 1 and Lemma 3.3 .