Global regular axially-symmetric solutions to the Navier-Stokes equations. Part 1

We prove existence of global regular axially-symmetric solutions to the Navier-Stokes equations in a cylindrical domain. We assume the periodic boundary conditions on the top and the bottom of the cylinder, but on the lateral part we assume vanishing of normal component of velocity, angular components of vorticity and angular component of velocity.


Introduction
In this paper we prove the existence of global regular axially-symmetric solutions to the Navier-Stokes equations in a cylindrical domain Ω ⊂ R 3 : where a, R are given positive numbers. We denote by x = (x 1 , x 2 , x 3 ) Cartesian coordinates. It is assumed that the x 3 -axis is the axis of symmetry of Ω. Moreover, where S 1 is parallel to the axis of symmetry and S 2 (a 0 ) is perpendicular to it. S 2 (a 0 ) meets the axis of symmetry at a 0 .
To describe the considered problem we introduce cylindrical coordinates r, ϕ, z by the relations (1.1) x 1 = r cos ϕ, x 2 = r sin ϕ, x 3 = z.
The cylindrical components of vorticity can be described in terms of the cylindrical components of velocity and swirl in the following form (1.13) Equation (1.7) 4 implies existence of the stream function ψ which is a solution to the problem (1.14) − ∆ψ + ψ r 2 = ω ϕ , ψ| S 1 = 0, ψ satisfies periodic boundary conditions on S 2 .
The aim of this paper is to prove the existence of global regular axially symmetric solutions to problem (1.6). For this purpose we have to find a global estimate guaranteeing the existence of global regular solutions.
Function ψ 1 is a solution to the problem We have that ω 1 = Γ.
To state the main result we first introduce assumptions.
In Lemma 4.5 the following quantity is defined Finally, in Lemma 4.7 we have introduced the quantity

The main result
Theorem 1.2. Assume that Assumption 1.1 holds. Then there exists an increasing positive function φ such that Remark 1.3. Estimate (1.24) implies any regularity of solutions to problem (1.6) assuming sufficient regularity of data.
To prove (1.24) we need that ψ 1 and v z vanish on the axis of symmetry. The proof of Theorem 1.2 is divided into the following steps: 1. In Lemmas 2.2 and 2.3 we prove the energy estimate for solutions to (1.6) and L ∞ -estimate for swirl.
2. In Lemma 2.5 the existence of weak solutions to problem (1.22) for the stream function ψ 1 is proved for a given ω 1 = ω/r. Solutions of (1.22) have the form ψ 1 = ψ/r. The weak solutions to (1.22) proved in Lemma 2.5 do not vanish on the axis of symmetry.
3. In Section 3 for a given ω 1 ∈ H 1 (Ω) many estimates for ψ 1 are found. In Lemma 3.3 we derived such estimate that ψ 1 must vanish on the axis of symmetry. We need the estimate in the proof of (1.24). The result of Lemma 3.3 shows that weak solutions proved in Lemma 2.5 must vanish on the axis of symmetry. In view of properties of the stream function it means that v z also vanishes on the axis of symmetry. In this section a theory of weighted Sobolev spaces from [NZ] is used.
4. In Section 4 we were able to derive an estimate for in terms of v ϕ L∞(0,t;L 12 (Ω)) and v ϕ ε 0 L∞(Ω t ) , where ε 0 can be chosen as small as we want. The estimate holds in view of Lemma 6.1 and inequality (2.12).
5. Finally, at the end of Section 4 and in Section 5 we were able to estimate v ϕ L∞(0,t;L 12 (Ω)) and v ϕ L∞(Ω t ) from the bound for (1.25).
The problem of regularity of axially-symmetric solutions to the Navier-Stokes equations has a long history. The first regularity results in the case of vanishing swirl are derived in [L2] and [UY] by O. A Ladyzhenskaya and Ukhovskii-Yudovich independently. Many references in the case of nonvanishing swirl can be found in [NZ1].
We have to emphasize that we were able to prove Theorem 1.2 because the theory of weighted Sobolev spaces developed in [NZ] was used.
Applying the Hölder inequality to the r.h.s. of (2.3) yields where f 2 = f 2 r + f 2 ϕ + f 2 z . Integrating (2.4) with respect to time gives Integrating (2.3) with respect to time, using the Hölder inequality in the r.h.s. of (2.3) and exploiting (2.5), we obtain The above inequality implies (2.1). This concludes the proof.
Integrating (2.9) with respect to time and passing with s → ∞, we derive (2.7). This ends the proof.
Proof. Multiplying (1.22) 1 by ψ 1 and using the boundary conditions we obtain Applying the Hölder and Young inequality to the r.h.s. implies (2.11). The Fredholm theorem gives existence. This ends the proof.
Remark 2.6. We have to emphasize that the weak solution ψ 1 of (1.22) does not vanish on the axis of symmetry. It also follows from [LW].
From Lemma 2.4 in [CFZ] we also have where f does not depend on ϕ.
Notation 2.8 (see [NZ]). First we introduce the Fourier transform. Let f ∈ S(R), where S(R) is the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on R. Then the Fourier transform of f and its inverse are defined by In view of transformation τ = − ln r, r = e −τ , dr = −e −τ dτ we have the equivalence In view of the Fourier tranform (2.13) and the Parseval identity we have 3 Estimates for the stream function ψ 1 Recall that ψ 1 is a solution to the problem (3.1) ψ 1 | r=R = 0, ψ 1 satisfies the periodic boundary conditions on S 2 .
Lemma 3.1. For sufficiently regular solutions to (3.1) the following estimates hold Proof. First we prove (3.2). Multiplying (3.1) 1 by ψ 1,zz and integrating over Ω yields Integrating by parts with respect to r in the first term implies Continuing, we get The first integral in (3.6) vanishes because ψ 1,r r| r=0 = 0, ψ 1,zz | r=R = 0. Integrating by parts with respect to z in the last term on the l.h.s. of (3.6) and using the periodic boundary conditions on S 2 we obtain Integrating by parts with respect to z in the first term in (3.7) and using the boundary conditions on S 2 we get Using this in (3.8) and applying the Hölder and Young inequalities to the r.h.s. of (3.8) yield Multiply (3.1) 1 by 1 r ψ 1,r and integrate over Ω. Then we have (3.10) The first term on the r.h.s. of (3.10) equals because ψ 1,r | r=0 = 0 (see [LW]). Applying the Hölder and Young inequalities to the last two terms on the r.h.s. of (3.10) implies (3.11) Inequalities (3.9) and (3.11) imply the estimate Inequalities (3.12) and (3.13) imply (3.2). Now, we show (3.3). Differentiate (3.1) 1 with respect to z, multiply by −ψ 1,zzz and integrate over Ω. Then, we obtain (3.14) Integrating by parts with respect to z yields where the first integral vanishes in view of periodic boundary conditions on S 2 . Integrating by parts with respect to r in the second integral in (3.15) gives where the first integral vanishes because In view of the above considerations, (3.14) takes the form Integrating by parts with respect to z in the last term on the l.h.s. of (3.16) and using the periodic boundary conditions on S 2 we get Applying the Hölder and Young inequalities to the r.h.s. of (3.17) yields where we used that ψ 1,zz | r=R = 0. The above inequality implies (3.3). Finally, we show (3.4). Differentiate (3.1) 1 with respect to z, multiply by ψ 1,rrz and integrate over Ω. Then we have Integrating by parts with respect to z in the second term in (3.18) implies where the first term vanishes because Then (3.18) takes the form Applying the Hölder and Young inequalities in the r.h.s. of (3.20) gives Inequalities ( Proof. Differentiating (3.1) with respect to z yields Using (3.27) in (3.32) implies (3.25). This concludes the proof.
Remark 3.4. Lemma 3.3 is necessary in the proof of global regular axiallysymmetric solutions to problem (1.6). However, it imposes strong restrictions on solutions to (1.6) because the condition ψ 1 | r=0 = 0 implies that v z | r=0 = 0. We do not know how to omit the restriction in the presented proof in this paper.
Lemma 3.5. Let µ > 0 and ω 1 ∈ H 1 µ (Ω). Then for sufficiently smooth solutions to (3.1) the following estimate is valid Proof. To prove the lemma we introduce a partition of unity where r 0 < R and ζ (i) (r), i = 1, 2, are smooth functions. Introduce the notation Then functions (3.34) satisfy the equations where dot denotes derivative with respect to r. First we consider the case i = 1. Differentiating (3.35) for i = 1 with respect to r yields ,r .
Let D 2 be defined by (2.7) and let
Multiplying (1.18) by Γ, integrating over Ω, using the boundary conditions and explanation about applying the Green theorem appeared below (4.2) we obtain (4.9) Using that Γ| r=R = 0, applying the Hölder and Young inequalities to the last term on the r.h.s. of (4.9) and using the Poincaré inequality we derive From (4.8) and (4.10) we have where φ is an increasing positive function. Integrating (4.11) with respect to time yields (4.1). This ends the proof.
Lemma 4.2. Let the assumptions of Lemma 6.1 hold.
where c 1 , c 2 depending on D 5 , D 7 are introduced in L 4 1 below.
Proof. We examine where ε = ε 1 + ε 2 and ε i , i = 1, 2, are positive numbers. Using (2.7) and applying the Hölder inequality in I 1 3 yields By the Hardy inequality we obtain Continuing, Now, we estimate the second factor L 2 1 .
Using (6.1) in L 3 1 , we have where c 1 , c 2 depend on D 5 , D 7 . To justify the above inequality we have to know that the following inequalities hold Consider (4.15). Using the form of q and s q we have 3 Hence Therefore the following inequality holds for d > 3 and ε 2 sufficiently small. Moreover, (4.17) implies To exmine (4.16) we calculate Since (4.19) must be positive we have the restriction Using (4.18) in (4.20) implies so there is no contradiction. Hence, we have where d > 3. Finally This implies (4.12) and ends the proof.
Remark 4.4. Consider exponents in (4.23). Then For ε 2 small we have where ε * , ε 0 * are positive number which can be chosen very small.
For d = 12 it follows that This ends the remark.
(4.34) The existence of global regular solutions to problem (1.6) for v ϕ sufficiently small is proved in Appendix B.
To derive any estimate from (4.39) we need We see that (4.40) holds for In view of the Young inequality, (4.39) implies The above inequality implies (4.27) and concludes the proof.

Estimates for the swirl
In this Section we find estimates for solutions to the problem (5.1) in Ω t , u| S 2 − periodic boundary conditions, u| t=0 = u(0) in Ω.
The third integral in (5.9) equals where the last term equals To examine the boundary term in J we recall the expansion of v ϕ near the axis of symmetry (see [LW]) v ϕ = a 1 (z, t)r + a 2 (z, t)r 3 + · · · , so u = a 1 (z, t)r 2 + a 2 (z, t)r 4 + · · · Then u ,rr + 1 r u ,r u ,r r| r=0 = 0 and we have to emphasize that all calculations in this paper are performed for sufficiently regular solutions.
Therefore, the boundary term in J equals Projecting (5.1) 1 on S 1 yields −ν u ,rr + 1 r u ,r + 2ν u ,r r = f 0 on S 1 . Hence Using the expression in J 1 gives The fourth term in (5.9) equals (5.12).
This inequality implies (5.3) and concludes the proof.