Global Regular Axially-Symmetric Solutions to the Navier–Stokes Equations with Small Swirl

Axially symmetric solutions to the Navier–Stokes equations in a bounded cylinder are considered. On the boundary the normal component of the velocity and the angular components of the velocity and vorticity are assumed to vanish. If the norm of the initial swirl is sufficiently small, then the regularity of axially symmetric, weak solutions is shown. The key tool is a new estimate for the stream function in certain weighted Sobolev spaces.


Introduction
In this work we consider axially-symmetric solutions to the Navier-Stokes equations in bounded cylindrical domains Ω ⊂ R 3 with the boundary S := ∂Ω.
The system of equations we investigate reads (1.1) where n is the unit outward normal to S vector.
To present our main result we need to introduce the quantity It is called the swirl and is a solution to the problem We have to emphasize that the boundary conditions (1.1) 3,4 were introduced by O.A. Ladyžhenskaya in [1].Condition (1.1) 4 is necessary for solvability of some initial-boundary value problems for ω ϕ (see (1.15) 2 ).
Consider now the case r 0 = R, thus Ω R = Ω.Suppose that Then One may wonder what is the difference between (1.5) and (1.6).Careful comparison shows that (1.5) is obtained provided that α(t, r 0 ) = u 2 L∞(Ω t r 0 ) is sufficiently small in the neighborhood of r = 0.In (1.6) we do not need any smallness restrictions.This might suggest that we can take r 0 = R and without any restrictions show the regularity of weak, axially symmetric solutions with non-vanishing v ϕ (0).Unsurprisingly, this is not true: (1.6) does not exist without obtaining (1.5) first.We will see later in the proof that we approach certain integral differently when r is close to 0 and when 0 < r 0 < r, where r 0 is fixed.Unfortunately, as (1.4) shows, passing with r 0 → 0 + is not possible.
We should emphasize that Theorem 1 does not directly imply the regularity of weak solutions but we may quickly deduce it following the reasoning from Lemma 2.9.Instead, we utilize one of many Serrin-type regularity criteria, e.g.[2, Theorem 3.(ii)], which states that if ω ϕ ∈ L ∞ (0, t; L 2 (Ω)), then a weak solution v to (1.1) is regular.Inequality (1.6) yields exactly ≤ cM ′ , and eventually In light of [3, Theorem 1] the above inequality also implies the regularity of a weak solution v to (1.1).In fact, there are many auxiliary results that could be utilized here.For a brief summary of Serrin-type regularity criteria for axially symmetric solutions to the Navier-Stokes equations we refer the reader to the introductions in e.g.[4], [5] and [6].Lots of regularity criteria in terms of angular component of the velocity or of the swirl were established in e.g.[7], [8], [9], [10], [11], [12], [13].
In general, the problem of regularity of weak solutions to the Navier-Stokes equations in R 3 has a long history.In 1968 it was shown independently by Ladyzhenskaya [1] and Ukhovskii et al. [14] that in class of axially symmetric solutions any weak solution is regular provided that v ϕ (0) = 0. Shortly after Ladyzhenskaya v2023-02-03 wrote a book ( [15]) which laid foundations for intensive research on regularity of weak solutions.
Before describing the steps of the proof of Theorem 1 let us briefly discuss recent results.In [16] the case Ω = R 3 is studied.Lei et al. show that if sup t≥0 u(r, z, t) ∼ O ln −2 r (see Corollary 1.3), then v is global and regular axially symmetric solution to (1.1) 1,2,6 .This is an improvement over Wei's result (see [17]), where O ln − 3 2 r is needed.These two results were recently improved in [18], where the condition u(r, z, t) ≤ N e −c|ln r| τ implies the regularity of weak solutions.Here 0 < r ≤ 1  4 and τ is any number from (0, 1), c, N are some constants.Our result is somehow comparable -(1.4) suggests that u(r, z, t) ∼ e − 1 r 16 .We have to emphasize that in papers [8], [10], [13] smallness condition looks very complicated and depends not only on the swirl but also on e.g.vorticity.
In [19] to prove the regularity of weak, axially symmetric solutions we assume either v r ∈ L ∞ (0, t; L 3 (Ω)) or vr r ∈ L ∞ (0, t; L 3 2 (Ω)).In both cases some smallness conditions are needed but they depend explicitly on the constant from the Poincaré inequality.
To the best of our knowledge that are not that many results concerning the regularity of weak, axially symmetric solutions to the Navier-Stokes equations in bounded cylinders (see e.g.[20]).Our main result is not only new but it also uses non-trivial weighted estimates for the stream functions.To explain this technique, we go back to (1.1) and following e.g.Ladyzhenskaya [1] or How et al. (see [21]) we rewrite it in the form (1.9) where F ϕ = rot f • ēϕ and ψ is the stream function such that We recall that in (1.9) and whenever cylindrical coordinates in this manuscript are used we have To derive energy type estimates for he velocity we prefer (1.1) 1,2 in the form (1.12) Moreover, we have the following boundary and initial conditions It is also convenient to introduce the quantities Then, system (1.9) finally reads Systems (1.15) and (1.9) are similar.Our main focus will be concentrated on To handle this integral we need estimates for solutions to both (1.15) and (1.9).These estimates are presented in Sections 2, 3 and 4. Finally, in Section 5 we eventually combine them.Apart from various energy estimates we also need two non-trivial estimates in weighted Sobolev spaces for solutions to (1.14) 3 (see Corollaries 2.10 and 2.11).Due to the order of the weight, we need to adjust the order of singularity of ψ 1 near r = 0.In Lemma 2.8 we will see that Section 2).Therefore, we subtract from ψ 1 as much as it is needed for this difference to belong to H 3 0 (Ω).This idea is motivated by Kondratev's work (see [22]) and discussed in a separate manuscript (see [23]).

First we introduce the function spaces
v2023-02-03 Definition 2.1.Let Ω be a cylindrical axially symmetric domain with axis of symmetry inside.We use the following notation for Lebesgue and Sobolev spaces: where s, k ∈ R + .
Finally, similarly to Definition 2.1 in [23] we introduce weighted spaces , In fact, we only use H 3 0 (Ω) and H 2 0 (Ω) and these symbols should not be mixed with Sobolev spaces with zero trace.
We use notation: r.h.s -right-hand side, l.h.s.-left-hand side.By c we denote generic constants.They are time-independent but they may depend on R. If a constant depends on a quantity l and this dependence needs to be tracked we write c(l).This means that c(l Lemma 2.2 (Hardy's inequality).Suppose that f ≥ 0, p ≥ 1 and r = 0.
Using that Applying the Hölder inequality to the r.h.s. of (2.3) yields where we used that |f | Integrating (2.3) with respect to time, using the Hölder inequality in the r.h.s. of (2.3) and using (2.5) we obtain . The above inequality implies (2.1) and concludes the proof.
3) 1 by u|u| s−2 , s > 2 integrating over Ω and by parts and using that u S = 0, we obtain where the last term of (2.Integrating (2.8) with respect to time and passing with s → ∞ we derive (2.6) from (2.8).This ends the proof.
Using weighted spaces we can estimate the r.h.s. of (2.12) by By the Hardy inequality (see Lemma 2.2) and µ ∈ (0, 1), r ≤ R, we get 11) holds.This concludes the proof.
Then for solutions to (1.15) the following inequality holds.
Applying the Hölder inequality to the r.h.s. of (2.14) and simplifying we get Integrating with respect to time yields Passing with s → ∞ we derive (2.13).This concludes the proof.
Lemma 2.8.Let ψ 1 be a solution to Suppose that ω 1 ∈ L 2 (Ω).Then, any solution ψ 1 to (2.17) satisfies Proof.We start with rewriting (2.17 Multiplying this equality by 1 r ψ 1,r and integrating over Ω yields The first term on the r.h.s. of (2.19) equals Integrating with respect to z in the second term on the r.h.s. of (2.19) yields where the first term vanishes because ψ 1,r | z∈{−a,a} = 0 and the second equals Using the boundary condition (2.17) 2 we obtain dz. v2023-02-03 From [25, Remark 4] we have Using (2.20) in I 1 yields Applying the Hölder and Young inequalities to the last term on the r.h.s in (2.19) and combining it with I 1 and I 2 we obtain Since the last two termns on the l.h.s. are positive we conclude that Now we can rewrite (2.17) in the form Lemma 2.9.Assume that s ∈ (1, ∞).Suppose that f ∈ L 1 (0, t; L s (Ω)) and u 1 (0) ∈ L s (Ω).Then Proof.In (2.14) we integrate by parts, use the boundary conditions (1.3) 4 and apply the Hölder and Young inequalities For sufficiently small ε we get , then c ǫ = cs 2 2(s−1)ν ≤ cs.Integrating with respect to time yields (2.27) we obtain (2.28) This concludes the proof.

Estimate for the angular component of velocity
Consider problem (1.9) ) Then, any solution to (1.9) satisfy Proof.Multiply (1.9) 1 by v 3 ϕ r 2 (see expansion (4.4) of v ϕ near the axis of symmetry) and integrate over Ω.Then we have The second term in (4.3) equals where we used that v • n S = 0 and div v = 0.
Integrating by parts in the third term on the l.h.s. of (4.3) yields The first term in I equals The middle term in J can be written in the form From [25,Remark 4] it follows that v ϕ behaves as for some functions a 1 and a 3 .Since v ϕ r=R = 0 the second terms in I and L vanish.
Using the above calculations in (4.3) yields Applying the Hölder and Young inequalities to the r.h.s. of (4.5) and integrating the result with respect to time imply (4.2).This concludes the proof.

Global estimate
Multiplying (3.1) by ν 2 4 and adding (4.2) we obtain Therefore, we have to estimate the first term on the r.h.s. of (5.1).To examine it we introduce the sets (5.2) where r 0 > 0 is given.We write the first term on the r.h.s. of (5.1) in the form (5.3) Lemma 5.1.Under the assumptions of Lemmas 2.3 and 2.5 we have Proof.Since vr r = −ψ 1,z we have In view of Lemma 2.5 the second term in J 1 is bounded by Note that all consideration are either a priori or performed for regular, local solutions.Then, derivation of regular, global solutions can be achieved by extension with respect to time.Since ψ is a solution to the problem we have Then J 2 is bounded by Using estimates for J 1 and J 2 we derive (5.4).This ends the proof.
Applying the Hardy inequality for the middle term in L 2 , gives where we used that f = −ψ 1,rr 1 + K(r) .To apply the Hardy inequality we use the formula Integrating the result with respect to z we derive the first inequality in L 3 .Continuing,  Exploiting the estimate in the bound of I we obtain (5.5).This concludes the proof.
It remains to check the convergence of X ′ n .Let Then, (5.10) implies (5.15) .
As explained after Theorem 1 we have to emphasize that (1.6) is crucial for deducing the regularity of weak solutions to problem (1.1). v2023-02-03