Abstract
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.
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1 Introduction
In this work we consider axially-symmetric solutions to the Navier–Stokes equations in bounded cylindrical domains \(\Omega \subset {\mathbb {R}}^{3}\) with the boundary \(S:=\partial \Omega \).
To describe the problem we transform the Cartesian coordinates \(x=(x_1,x_2,x_3)\) into cylindrical coordinates by the relation
This relation determines the orthonormal basis \(({\bar{e}}_r, \bar{e}_\varphi , {\bar{e}}_z)\), where
are unit vectors along the radial-, the angular-, and the z-axes, respectively.
Using this orthonormal basis we can decompose the velocity vector \(\textbf{v}\) as follows
For the vorticity vector \(\varvec{\omega }=\textrm{rot}\,\textbf{v}\) we have the expression
Here \(\omega _\varphi \) can be computed explicitly, i.e. \(\omega _\varphi =v_{r,z}-v_{z,r}\).
Let \(R, a > 0\). Then, we define
and by \(\partial \Omega = S_1\cup S_2\) we denote the boundary of \(\Omega \), where
The system of equations we investigate reads
where \({\bar{n}}\) 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 Ladyžhenskaya in [1]. Condition (1.1)\(_4\) is necessary for solvability of some initial-boundary value problems for \(\omega _\varphi \) (see (1.15)\(_2\)).
Theorem 1
(Main result). Fix \(0< r_0 < R\). Let
Let us introduce
Let
Assume that \(\gamma > 1\) and \(\alpha (t, r_0)\) is so small that
Then
Consider now the case \(r_0=R\), thus \(\Omega _R=\Omega \). 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 \(\alpha (t,r_0) = \left\Vert u\right\Vert ^2_{L_\infty (\Omega _{r_0}^t)}\) 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 \(v_\varphi (0)\). 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 \(r_0 \rightarrow 0^+\) 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 \(\omega _\varphi \in L_\infty (0,t;L_2(\Omega ))\), then a weak solution \(\textbf{v}\) to (1.1) is regular. Inequality (1.6) yields exactly
which for \(\textbf{v}'=(v_r,v_z)\) yields
and eventually
In light of [3, Theorem 1] the above inequality also implies the regularity of a weak solution \(\textbf{v}\) 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 \({\mathbb {R}}^{3}\) 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 \(v_\varphi (0) = 0\). Shortly after Ladyzhenskaya 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 \(\Omega = {\mathbb {R}}^{3}\) is studied. Lei et al. show that if \(\sup _{t \ge 0} \left|u(r,z,t)\right| \sim O\left( \ln ^{-2} r\right) \) (see Corollary 1.3), then \(\textbf{v}\) is global and regular axially symmetric solution to (1.1)\(_{1,2,6}\). This is an improvement over Wei’s result (see [17]), where \(O\left( \ln ^{-\frac{3}{2}} r\right) \) is needed. These two results were recently improved in [18], where the condition
implies the regularity of weak solutions. Here \(0 < r \le \frac{1}{4}\) and \(\tau \) is any number from (0, 1), c, N are some constants. Our result is somehow comparable—(1.4) suggests that \(\left|u(r,z,t)\right| \sim e^{-\frac{1}{r^{16}}}\).
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 \(v_r \in L_\infty (0,t;L_3(\Omega ))\) or \(\frac{v_r}{r} \in L_\infty (0,t; L_{\frac{3}{2}}(\Omega ))\). 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
where \(F_\varphi =\textrm{rot}\,\textbf{f}\cdot {\bar{e}}_\varphi \) and \(\psi \) 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 the velocity we prefer (1.1)\(_{1,2}\) in the form
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 \(\int _{\Omega ^t} \frac{v_r}{r} \frac{v_\varphi ^4}{r^2}\, \textrm{d}x\textrm{d}t'\). To handle this integral we need estimates for solutions to both (1.15) and (1.9). These estimates are presented in Sects. 2, 3 and 4. Finally, in Sect. 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 \(\psi _1\) near \(r = 0\). In Lemma 2.8 we will see that \(\psi _1 \sim O(1)\), thus \(\psi _1 \notin H^3_0(\Omega )\) (see Sect. 2). Therefore, we subtract from \(\psi _1\) as much as it is needed for this difference to belong to \(H^3_0(\Omega )\). This idea is motivated by Kondratev’s work (see [22]) and discussed in a separate manuscript (see [23]).
2 Notation and Auxiliary Results
First we introduce the function spaces
Definition 2.1
Let \(\Omega \) be a cylindrical axially symmetric domain with axis of symmetry inside. We use the following notation for Lebesgue and Sobolev spaces:
where \(p,q\in [1,\infty ]\), Q replaces either \(\Omega \) or S.
where \(s,k\in {\mathbb {R}}^{1}_+\).
Finally, similarly to Definition 2.1 in [23] we introduce weighted spaces \(L_{p,\mu }(\Omega )\), \(\mu \in {\mathbb {R}}^{1}\), \(p\in [1,\infty ]\), with the norm
and
where \(\textrm{D}^\alpha _{r,z} = \partial _r^{\alpha _1}\partial _z^{\alpha _2}\), \(\left|\alpha \right| = \alpha _1 + \alpha _2\), \(\left|\alpha \right| \le k\), \(\alpha _i \in {\mathbb {N}}_0 \equiv \{0,1,2,\ldots \}\), \(i=1,2\), \(k\in {\mathbb {N}}_0\) and \(\mu \in {\mathbb {R}}\). In fact, we only use \(H^3_0(\Omega )\) and \(H^2_0(\Omega )\) 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 \(c(l) \sim c \cdot l\). Similarly \(c\left( \frac{1}{l}\right) \sim \frac{c}{l}\).
Lemma 2.2
(Hardy’s inequality). Suppose that \(f\ge 0\), \(p\ge 1\) and \(r\ne 0\). Then
Lemma 2.3
Let \(\textbf{f}\in L_{2,1}(\Omega ^t)\), \(\textbf{v}(0)\in L_2(\Omega )\). Assume that \(v_\varphi \big \vert _S=0\), \({\bar{n}}\cdot \textbf{v}\big \vert _S=0\), \(\omega _\varphi \big \vert _S=0\). Then, solutions to (1.1) satisfy the estimate
Proof
Multiplying (1.12)\(_1\) by \(v_r\), (1.12)\(_2\) by \(v_\varphi \), (1.12)\(_3\) by \(v_z\), adding the results, integrating over \(\Omega \) and using (1.13) yields
In view (1.13) the boundary terms in (2.2) vanish. The last term on the l.h.s. of (2.2) vanishes in virtue of (1.13) and the equation of continuity (1.12)\(_4\).
Using that \(\left|\textbf{v}\right|^2=v_r^2+v_\varphi ^2+v_z^2\), we rewrite (2.2) the form
Applying the Hölder inequality to the r.h.s. of (2.3) yields
where we used that \(\left|\textbf{f}\right|^2=f_r^2+f_\varphi ^2+f_z^2\).
Integrating (2.4) with respect to time implies
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. \(\square \)
Lemma 2.4
Consider problem (1.3). Assume that \(f_0\in L_{\infty ,1}(\Omega ^t)\), \(u(0)\in L_\infty (\Omega )\). Then
Proof
Multiplying (1.3)\(_1\) by \(u|u|^{s-2}\), \(s>2\) integrating over \(\Omega \) and by parts and using that \(u\big \vert _S=0\), we obtain
where the last term of (2.7) equals \(I \equiv \frac{\nu }{s} \int _{-a}^a \left|u\right|^s\big \vert _{r=0}^{r=R}\, \textrm{d}z\). From [24] it follows that \(u\big \vert _{r=0}=0\). Since \(u\big \vert _{r=R}=0\) and using the boundary condition \(v_\varphi \big \vert _{S} = 0\) we conclude that \(I = 0\). Then, we derive from (2.7) the inequality
Integrating (2.8) with respect to time and passing with \(s\rightarrow \infty \) we derive (2.6) from (2.8). This ends the proof. \(\square \)
Lemma 2.5
Let estimates (2.1) and (2.6) hold. Then
Proof
We have
This implies (2.9) and concludes the proof. \(\square \)
Lemma 2.6
Let \(\omega _1\in L_2(\Omega )\). Then solutions to (1.15)\(_3\) satisfy
where \(\psi _1(0)=\psi _1|_{r=0}\). In addition, if \(\omega _1\in L_{2,\mu }(\Omega )\), \(\mu \in (0,1)\) then
where \(\psi _1(0)=\psi _1|_{r=0}\).
Proof
Multiply (1.15)\(_3\) by \(\psi _1\), integrate over \(\Omega \) and use boundary condition (1.15)\(_4\). Then we obtain
Applying the Hölder inequality to the r.h.s. of (2.12), using the Poincaré inequality and boundary condition (1.3)\(_4\) we obtain (2.10).
Using weighted spaces we can estimate the r.h.s. of (2.12) by
By the Hardy inequality (see Lemma 2.2) and \(\mu \in (0,1)\), \(r\le R\), we get
Since \(\mu \in (0,1)\) the bound \(\int _\Omega \left|\psi _1\right|^2 r^{-2\mu }\, \textrm{d}x < \infty \) does not imply \(\psi _1\big \vert _{r=0} = 0\). Then (2.11) holds. This concludes the proof. \(\square \)
Lemma 2.7
Assume that \(u_1(0)\in L_\infty (\Omega )\), \(f_1, \psi _{1,z}\in L_1(0,t;L_\infty (\Omega ))\). Then for solutions to (1.15) the following inequality
holds.
Proof
Multiply (1.15)\(_1\) by \(u_1|u_1|^{s-2}\) and integrate over \(\Omega \). Then we have
Applying the Hölder inequality to the r.h.s. of (2.14) and simplifying we get
Integrating with respect to time yields
Passing with \(s\rightarrow \infty \) we derive (2.13). This concludes the proof. \(\square \)
Lemma 2.8
Let \(\psi _1\) be a solution to
Suppose that \(\omega _1 \in L_2(\Omega )\). Then, any solution \(\psi _1\) to (2.17) satisfies
Proof
We start with rewriting (2.17)\(_1\) in the form
Multiplying this equality by \(\frac{1}{r}\psi _{1,r}\) and integrating over \(\Omega \) 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 \(\psi _{1,r}\vert _{z \in \{-a,a\}} = 0\) and the second equals
Using the boundary condition (2.17)\(_2\) we obtain
From [25, Remark 4] we have
thus
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
and consider it as the Dirichlet problem for the Poisson equation. Thus
where (2.22) was used. This ends the proof. \(\square \)
Lemma 2.9
Assume that \(s \in (1,\infty )\). Suppose that \(f\in L_1(0,t;L_s(\Omega ))\) and \(u_1(0) \in L_s(\Omega )\). 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 \(\varepsilon \) we get
Hence, we have
Since \(\epsilon = \frac{2(s-1)\nu }{s^2}\), then \(\frac{c}{\epsilon } = \frac{cs^2}{2(s-1)\nu } \le cs\). Integrating with respect to time yields
Using Lemma 2.8
we obtain
This concludes the proof. \(\square \)
Corollary 2.10
(Theorem 1.3 in [23]). Suppose that \(\psi _1\) is a weak solution to (1.15)\(_{3,4}\). Let \(\omega _1\in L_2(\Omega )\) and introduce
where \(K(\tau )\) is any smooth function with a compact support such that
Then
Corollary 2.11
(Theorem 1.4 in [23]). Let \(\psi _1\) be a weak solution to (1.3)\(_{3,4}\). Let \(\omega _1\in H^1(\Omega )\). Then
where
and K is the same as in Corollary 2.10.
3 Estimate for \(\omega _1\)
Lemma 3.1
Assume that \(\omega _1(0)\in L_2(\Omega )\), \(u_1\in L_4(\Omega ^t)\), \(F\in L_{6/5,2}(\Omega ^t)\), \(t\le T\). Then the following inequality holds
Proof
Multiply (1.15)\(_2\) by \(\omega _1\), integrate over \(\Omega \), integrate by parts. Next, integration with respect to time implies (3.1). This ends the proof. \(\square \)
4 Estimate for the Angular Component of Velocity
Consider problem (1.9)
Lemma 4.1
Assume that \(f_\varphi \in L_2(\Omega ^t)\), \(v_\varphi (0)\in L_{4,-1/2}(\Omega )\),
Then, any solution to (1.9) satisfy
Proof
Multiply (1.9)\(_1\) by \({v_\varphi ^3\over r^2}\) (see expansion (4.4) of \(v_\varphi \) near the axis of symmetry) and integrate over \(\Omega \). Then we have
The second term in (4.3) equals
where we used that \(\textbf{v}\cdot {\bar{n}}\big \vert _S=0\) and \({{\,\textrm{div}\,}}\textbf{v} = 0\).
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 \(v_\varphi \) behaves as
for some functions \(a_1\) and \(a_3\). Since \(v_\varphi \big \vert _{r=R}=0\) 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. \(\square \)
5 Global Estimate
Multiplying (3.1) by \(\frac{\nu ^2}{4}\) 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
where \(r_0>0\) is given.
We write the first term on the r.h.s. of (5.1) in the form
Lemma 5.1
Under the assumptions of Lemmas 2.3 and 2.5 we have
Proof
Since \({v_r\over r}=-\psi _{1,z}\) 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 \(\psi \) 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. \(\square \)
Lemma 5.2
Let the assumptions of Lemma 2.3 hold. Additionally, assume that \(v_\varphi (0)\in L_4(\Omega )\), \(u\in L_\infty (\Omega ^t)\), \(\left|u\right|_{\infty ,\Omega ^t}\le D_2\). Then I from (5.3) satisfies
Proof
We have
We estimate \(I_2\) by
where \(\psi _1(0)=\psi _1\big \vert _{r=0}\) and \(\eta \) is defined in Corollary 2.11. Using this Corollary we have
Consider \(I_2^3\),
where we used Lemmas 2.3 and 2.8.
Consider \(I_2^2\). To simplify presentation we express \(\eta \) in the short form
where f replaces \(\left( \frac{3}{r}\psi _{1,r} + \psi _{1,zz} + \omega _1\right) \left( 1 + K(r)\right) \).
Then
Using the Hölder inequality in \(L_1\) implies
where \(1/p+1/p'=1\).
Applying the Hardy inequality for the middle term in \(L_2\), gives
where we used that
To apply the Hardy inequality we use the formula
Then, we use the following Hardy inequality (see e.g. [26, Ch. 1, Sec. 2.16])
Integrating the result with respect to z we derive the first inequality in \(L_3\). Continuing,
Using
in \(L_4\) implies
where the properties of K are used. Finally, for \(p\le 3\) and Lemma 2.8
Summarizing
where \(p'\ge \frac{3}{2}\) and Lemma 2.8 was used.
Using estimates of \(I_2^1\), \(I_2^2\), \(I_2^3\), we obtain
Exploiting the estimate in the bound of I we obtain (5.5). This concludes the proof. \(\square \)
Proof
(r) Using (5.3) and estimates (5.4) and (5.5) in (5.1) and assuming that \(\varepsilon _1\) and \(\varepsilon _3\) are sufficiently small we obtain the inequality
where M(t) is introduced in (1.4).
Let
In view of this notation, (5.6) takes the form
Consider the equality
Using the method of successive approximations we will show that there exists a solution \(X'(t)\) and determine the magnitude of \(\epsilon \) which ensures the existence of this solutions.
Suppose that
Let \(\gamma > 1\). Recall that \(M = M(T)\) and assume that
Then from (5.10) and (5.8) we obtain
Assume that
Then
Let now \(\omega _1(0)\) be given. Let \(\tilde{\omega }_1\) be an extension of \(\omega _1(0)\) such that \(\left|\tilde{\omega }_1\right|_{2,\infty (\Omega ^t)}^2 + \left\Vert \tilde{\omega }_1\right\Vert _{1,2,\Omega ^t}^2 < \infty \) and \(\tilde{\omega }_1\big \vert _{t = 0} = \omega _1(0)\). Let
Then, (5.11), (5.13) and (5.14) imply that
It remains to check the convergence of \(X_n'\). Let
Then, (5.10) implies
Continuing, we have
Hence, the sequence converges if
This ends the proof. \(\square \)
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).
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Nowakowski, B., Zajaczkowski, W.M. Global Regular Axially-Symmetric Solutions to the Navier–Stokes Equations with Small Swirl. J. Math. Fluid Mech. 25, 73 (2023). https://doi.org/10.1007/s00021-023-00793-9
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DOI: https://doi.org/10.1007/s00021-023-00793-9