Abstract
We point out some criteria that imply regularity of axisymmetric solutions to Navier–Stokes equations. We show that boundedness of \(\Vert {v_{r}}/{\sqrt{r^3}}\Vert _{L_2({\mathbb {R}}^3\times (0,T))}\) as well as boundedness of \(\Vert {\omega _{\varphi }}/{\sqrt{r}} \Vert _{L_2({\mathbb {R}}^3\times (0,T))}\), where \(v_r\) is the radial component of velocity and \(\omega _{\varphi }\) is the angular component of vorticity, imply regularity of weak solutions.
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1 Introduction
We consider the Cauchy problem to the three-dimensional axisymmetric Navier–Stokes equations:
where \(x= (x_1, x_2, x_3),\ v\) is the velocity of the fluid motion with
\(p=p(x,t)\in {\mathbb {R}}^1\) denotes the pressure, \(\nu \) is the viscosity coefficient 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 [1] and Yudovich–Ukhovskij [2] 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 [3,4,5,6]. 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 [7]).
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 [6,7,8,9,10,11,12] and the literature cited in these papers). In Sect. 2 we recall only such criteria that are useful for our analysis.
Since we are restricted to the axisymmetric solutions we introduce the cylindrical coordinates \((r,\varphi ,z)\) by the relations
and corresponding unit vectors:
Then the cylindrical components of velocity and vorticity (\(\omega =\mathrm{rot}\,v\)) for axisymmetric solutions (therefore, solutions independent of \(\varphi \)) are represented as
and
where \(v_r, v_\varphi , v_z\) are radial, angular and axial components of velocity.
The axisymmetric motion can be described by the three quantities: \(v_{\varphi }, \omega _{\varphi }\) and the stream potential \(\psi \) which are solutions to the following equations:
where \(v\cdot \nabla = v_r\partial _r + v_z\partial _z, \Delta =\partial _r^2 + \partial _z^2 + \frac{1}{r}\partial _r \) and
It is very convenient to introduce quantities \(u_1, \omega _1, \psi _1\) by the relations
Then Eqs. (1.2) simplify to
and
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 Eqs. (1.1) with the axisymmetric initial data and additionally \(\mathrm{div}\,v(0) = 0\).
-
2.
Assume that \(\displaystyle {\frac{v^2_{\varphi }(0)}{r}, \frac{\omega _{\varphi }(0)}{r}, \frac{\omega _r(0)}{r}, \omega _z(0)}\) belong to \(L_2(R^3)\), and with the notation \(u=rv_{\varphi }, u(0)\in L_{\infty }({\mathbb {R}}^3)\bigcap L_s({\mathbb {R}}^3), s\ge 3.\)
-
3.
Assume that there exists constant \(c_1\) such that
$$\begin{aligned} \int _0^T {\textit{dt}} \int _{{\mathbb {R}}^3}\frac{v_r^2}{r^3} {\textit{dx}} \le c_1 < \infty . \end{aligned}$$(1.7)Then \(v \in L_{\infty }(0,T;H^1({\mathbb {R}}^3_{r_0})),\) where \({\mathbb {R}}^3_{r_0}= \{x\in {\mathbb {R}}^3, r<r_0 \}\) and \(r_0>0\) is given. Assume additionally that \(v(0)\in B^{2-2/r}_{\sigma ,r}({\mathbb {R}}^3_{r_0})\) - Besov space. Then \(v\in W^{2,1}_{\sigma ,r}({\mathbb {R}}^3_{r_0}\times (0,T))\).
Remark 1.1
For \(\sigma > 3, r=2\) we have that \(v\in L_{\infty }({\mathbb {R}}^3_{r_0} \times (0,T))\) so in view of [13] there is no singular points. In \({\bar{{\mathbb {R}}}}^3_{r_0}= \{x\in {\mathbb {R}}^3, r> r_0 \}\) 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
then there exists a constant \(c_3\) such that
2 Notation and Auxiliary Results
By \(L_p({\mathbb {R}}^3), p\in [1,\infty ],\) we denote the Lebesgue space of integrable functions. By \(L_{p,q}({\mathbb {R}}^3 \times (0,T))\) we denote the anisotropic Lebesgue space with the following finite norm
where \(p,q \in [1,\infty ].\)
We define Sobolev spaces \(W_p^{2,1}({\mathbb {R}}^3 \times (0,T))\) and \(W^{2-2/p}_p({\mathbb {R}}^3 \times (0,T))\) by
where [l] is the integer part of l.
By \(H^s({\mathbb {R}}^3), s\in {\mathbb {N}}_0 = {\mathbb {N}}\cup \{0\}\) we denote the Sobolev space \(W^s_2({\mathbb {R}}^3)\).
To find the form of \(|\nabla v|\) we recall that
Using that \(v = v_r {\bar{e}}_r + v_{\varphi } {\bar{e}}_{\varphi } + v_z {\bar{e}}_z\), we obtain
where we used the fact that vectors \({\bar{e}}_r, {\bar{e}}_{\varphi }, {\bar{e}}_z\) are orthonormal.
Lemma 2.1
There exists a weak solution to problem (1.1) such that \(v \in L_{\infty }(0,T;L_2({\mathbb {R}}^3))\cap L_2(0,T;H^1({\mathbb {R}}^3))\) and the following estimate holds
In the case of axisymmetric solutions the energy inequality (2.1) takes the form
Proof
Equations \((1.1)_{1,2}\) for the axially symmetric solutions assume the form
where \(v\cdot \nabla =v_r\partial _r+v_z\partial _z\), \(\Delta u=\frac{1}{r}(ru_{,r})_{,r}+u_{,zz}\).
Let \(I= \{ (\varphi , r, z): r=0\}\) denote the axis of symmetry. Any space of functions which vanish on I considered in this paper is a closure of \(C^{\infty }_0({\mathbb {R}}^3{\setminus } I)\) in the corresponding norm. Then, we are looking for a priori estimate for functions v having such property.
Multiplying (2.4) by \(v_\varphi \) and integrating over \({\mathbb {R}}^3\) yields
Multiplying (2.3) by \(v_r\), integrating over \({\mathbb {R}}^3\) implies
Multiplying (2.5) by \(v_z\) and integrating over \({\mathbb {R}}^3\) we obtain
Adding the above equations and using (2.6) we obtain
Integrating (2.7) with respect to time from 0 to t, \(t\le T\), yields
This ends the proof. \(\square \)
To derive energy estimates in the proof of Lemma 2.1 we use the ideas from the proof of Theorem 3.1 from [14, Ch.3]. The notion of a suitable weak solution was introduced by Caffarelli et al. [13] in the famous paper. 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 \(v(0)\in W_p^{2-2/p}({\mathbb {R}}^3)\) then \(v\in W^{2,1}_p({\mathbb {R}}^3 \times R_+)\) for any \(p\in (1,\infty )\). Hence for \(p>\frac{5}{2}\) we have that \(v\in L_{\infty }({\mathbb {R}}^3 \times R_+)\) so it is also bounded locally. Therefore v has no singular points (see [13]). To show this we use results of J. Neustupa, M. Pokorny and O. Kreml (see [5, 6, 10]). To clarify presentation we recall the results.
From [5] it follows that
Lemma 2.2
[5]. Let v be an axisymmetric suitable weak solution to problem (1.1). Suppose that there exists a subdomain \(D \subset {\mathbb {R}}^3\times {\mathbb {R}}_+\) such that the angular component \(v_{\varphi }\) of v belongs to \(L_{s,r}(D)\) where
- 1.
Either \(s\in [6,\infty ], r\in [20/7, \infty ]\) and \(2/r+3/s\le 7/10\).
- 2.
or \(s\in [24/5,6], r\in [10,\infty ]\) and \(2/r+3/s\le 1-9/(5s)\).
Then v has no singular points in D.
Lemma 2.3
[10]. Let v be an axisymmetric suitable weak solution to problem (1.1). Suppose that there exists a subdomain \(D \subset {\mathbb {R}}^3\times {\mathbb {R}}_+\) such that the angular component \(v_{\varphi }\) of v belongs to \(L_{s,r}(D)\) where
Then v has no singular points in D.
By swirl we denote
From (1.2)\(_1\) it follows that u satisfies the equation
Lemma 2.4
(See [3]). Let \(u(0)= r v_{\varphi }(0) \in L_{\infty }({\mathbb {R}}^3).\) Then
Remark 2.5
We recall also Lemma 3.1 from [9]
Lemma 2.6
(See [9]). Assume that (v, p) is regular axisymmetric solution to the Navier–Stokes Eqs. (1.1). Assume that \(\frac{v_{\varphi }(0)}{\sqrt{r}}\in L_4({\mathbb {R}}^3), {\tilde{\nabla }}\frac{v_r}{r} \in L_{4/3}(0,T; L_2({\mathbb {R}}^3))\) when \({\tilde{\nabla }}=(\partial _r, \partial _z).\) Then the following estimate holds
Proof
Consider the following problem for \(v_{\varphi }\) in \({\mathbb {R}}^3_{\varepsilon }= \{x\in {\mathbb {R}}^3: r>\varepsilon \}, \varepsilon >0,\)
Multiplying (2.11) by \(\frac{v_{\varphi }^3}{r^2},\) integrating over \({\mathbb {R}}^3_{\varepsilon }\) and using boundary conditions yields
Simplifying we have
By the Gronwall lemma we have
Passing with \(\varepsilon \rightarrow 0\) we derive (2.10) and conclude the proof. \(\square \)
Remark 2.7
Formula (2.4) in [9] has the form
Consider problem (1.3).
Lemma 2.8
Let the assumptions of Lemma 2.6 be satisfied. Assume additionally that \(w_1(0) \in L_2({\mathbb {R}}^3), u_1(0) \in L_4(0,t;L_4({\mathbb {R}}^3)).\) Then the following estimate holds
Proof
Consider the problem in \({\mathbb {R}}^3_{\varepsilon }\)
Multiplying (2.14) by \(\omega _1\), integrating over \({\mathbb {R}}^3_{\varepsilon }\), using the boundary conditions yields
Integrating with respect to time and passing with \(\varepsilon \rightarrow 0\) gives (2.13). This concludes the proof. \(\square \)
Introduce the quantities
which are solutions to the equations
Remark 2.9
In the proof of Theorem 1.1, Case 1 in [9] there is derived the formula (3.8) in [9] in the form
where
Let us recall some properties of weak solutions to (1.1).
Lemma 2.10
(See [15, Ch.2, Sect. 3]). For arbitrary \(v\in L_{\infty }(0,T;L_2(\Omega ) \, \cap L_2(0,T;H^1(\Omega ))\) the inequality holds:
where
3 Sufficient Conditions for Regularity
Let
Then \(u_{\alpha }\) satisfies
Lemma 3.1
Let \(u(0) \in L_{\infty }({\mathbb {R}}^3)\). Assume that \(c_1\) is a constant and
Let
Then
Proof
Multiplying (3.1) by \(u_{\alpha }|u_{\alpha }|^{s-2},\) integrating the result over \({\mathbb {R}}^3\) and using that \(u_{\alpha } \in C^{\infty }({\mathbb {R}}^3\setminus I)\), and \(u_{\alpha }\) has compact support for large, finite \(x\in {\mathbb {R}}^3,\) we obtain
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 \(\varepsilon =\nu \alpha (2-\alpha ),\) integrating the result with respect to time and using the density argument, yields
In view of the assumptions of the lemma, the r.h.s. of (3.3) is bounded by a constant \(c_2\). Then the density argument and Lemma 2.10 imply
Hence
where
Comparing the above approach with (2.15) we have that \(d=1-\alpha \) and the regularity criterion has the form
Hence
Therefore \(\alpha s=3\). \(\square \)
Corollary 3.2
From (3.2), (2.15) and (2.10) we obtain
where \(c_3\) depends on the constants from the r.h.s. of (3.2), (2.15) and (2.10).
In view of (2.16) we have
where
Let \({\mathbb {R}}^3_{r_0} = \{x\in {\mathbb {R}}^3: r\le r_0 \}.\) Then (3.4) implies
Hence (3.5) implies that \(v_{\varphi } \in L_{p',q'}({\mathbb {R}}^3_{r_0}\times (0,T)),\) where
Consider Lemma 2.3. Let \(s=4\). Then \(r=8\), so
Since \(\frac{3}{4} < 1, v_{\varphi }\) satisfies assumptions of Lemma 2.3. Hence v has no singular points in \({\mathbb {R}}^3_{r_0}\times (0,T).\)
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 [13] 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.
Consider Eq. (1.2)\(_1\). Let \(\chi = \chi (r)\) be a smooth function such that
We multiply (1.2)\(_1\) by \(\chi \) and introduce the notation \( {\hat{v}}_{\varphi } = v_{\varphi } \chi \). Then \({\hat{v}}_{\varphi }\) satisfies
Multiplying (3.6) by \({\hat{v}}_{\varphi }|{\hat{v}}_{\varphi }|^{s-2}\) and integrating over \({\mathbb {R}}^3\times (0,t)\) yields
In view of Lemma 2.10 the first two terms on the l.h.s. of (3.7) are estimated from below by
Next we estimate non-positive terms in (3.7). In view of the energy estimate (2.1) the last term on the l.h.s. of (3.7) is bounded by
where the second factor can be always absorbed by (3.8) because \(5/3 >10/7.\) In view of (2.9) and (2.1) the first integral on the r.h.s. of (3.7) is bounded by
The last factor in (3.9) can be absorbed by (3.8) for \(\frac{20}{9}(s-1) \le \frac{5}{3}s\) which holds for \(s\le 4.\) Then (3.8) yields estimate for
Finally we estimate the last two integrals on the r.h.s. of (3.7). We write them in the form
where
and the first factor is bounded in view of Remark 2.5 and the second factor is absorbed by (3.8) for \(2(s-2)\le \frac{5}{2} s\) which holds for \(s\le 12.\) 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 \(2(s-2)\le \frac{5}{3}s\) so \(s\le 12.\) Summarizing, we obtain the estimate
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 \(s\ge \frac{50}{7}.\)
To increase regularity we introduce a new smooth cut-off function
Introducing notation \( \breve{v}_{\varphi }= {v}_{\varphi }\breve{\chi }\) we replace (3.6) by
where we can use (3.10). Hence the above problem increases regularity of (3.10), so we can meet assumptions of Lemma 2.2(1). This concludes the proof.
Lemma 3.3
Assume that
Then there exists a constant c such that
Proof
From (1.2)\(_3\) we have
where \(v_r= -\psi _{,z}.\)
To simplify further considerations we introduce the notation
Then (3.12) takes the form
Recall that I is the axis of symmetry. Let \(C^{\infty }_0({\mathbb {R}}^3{\setminus } I)\) be the set of smooth functions vanishing near I and outside a compact set.
Let \(H^1_0({\mathbb {R}}^3)\) be the closure of \(C^{\infty }_0({\mathbb {R}}^3{\setminus }I)\) in the norm
Recall that functions from \(H^1_0({\mathbb {R}}^3)\) vanish on I. By the weak solution to (3.13) we mean a function \(u\in H^1_0({\mathbb {R}}^3)\) satisfying the integral identity
which holds for any smooth function \(\chi \) belonging to \(H^1_0({\mathbb {R}}^3).\) Introducing the scalar product
we can write (3.14) in the following short form
where
For \(f\in L_2({\mathbb {R}}^3)\), \( (f, \chi _{,z})_{L_2({\mathbb {R}}^3)}\) is a linear functional on \(H^1_0({\mathbb {R}}^3).\) Hence we have
Hence by the Riesz Theorem there exists \(F\in H^1_0({\mathbb {R}}^3)\) such that
Therefore there exists a solution to the integral identity (3.15) such that \(u=F\in H^1_0({\mathbb {R}}^3)\) and the estimate holds
It is clear that the solution is unique.
Since u vanishes for \(r=0\), \(u=\psi _{,z}\) so \(\psi |_{r=0}=0\) also. Therefore we can look for approximate weak solution satisfying the integral identity
where \({\mathbb {R}}^3_{\varepsilon }= \{x\in {\mathbb {R}}^3: r>\varepsilon \}, \varepsilon >0.\) Let \(\chi =\frac{\psi _{,z}}{r^{\alpha }}.\) Then recalling notation \(u=\psi _{,z}, f=\omega _{\varphi }\) identity (3.16) takes the form
Performing differentiation we have
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 \(\psi \) in (3.18) is an approximate function. This should be denoted with \(\psi ^{\varepsilon }\) but we omitted it for simplicity.
Passing with \(\varepsilon \rightarrow 0\), setting \(\alpha =1\) and integrating this inequality with respect to time implies (3.11). This concludes the proof. \(\square \)
Proof of Theorem 1
Proof
From (2.10) and (2.12) we have
Next (2.15) implies
Finally, in view of (1.7) and Lemma 3.1 we get
where \(\alpha =\frac{3}{s}, s\ge 3, t\le T.\) These estimates imply that
where h is some positive increasing function of its arguments. Hence Lemma 2.3 implies local regularity.
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 [5] that
where data are data from the assumptions of the Step 1 of the proof of Theorem 1 from [5].
Considering the problem
and the local technique from [15, Ch.4, Sect.10], we have
Consider the problem
Employing the result of Solonnikov, estimate for v above and some interpolation we get
where \(\sigma <6, r\) arbitrary. This proves the second part of Theorem 1. \(\square \)
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Rencławowicz, J., Zaja̧czkowski, W.M. On Some Regularity Criteria for Axisymmetric Navier–Stokes Equations. J. Math. Fluid Mech. 21, 51 (2019). https://doi.org/10.1007/s00021-019-0447-0
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DOI: https://doi.org/10.1007/s00021-019-0447-0