A Note on Local Regularity of Axisymmetric Solutions to the Navier–Stokes Equations

In the paper, a new slightly supercritical condition, providing local regularity of axially symmetric solutions to the non-stationary 3D Navier-Stokes equations, is discussed. It generalises almost all known results in the local regularity theory of weak axisymmetric solutions.


Introduction
In the submitted note, we study potential singularities of axially symmetric flows of viscous incompressible fluids. It has been shown in the paper [17] that an axisymmetric solution to the three-dimensional nonstationary Navier-Stokes equations with a bounded scale-invariant energy quantity is actually smooth, i.e., axially symmetric solutions do no exhibit Type I blowups (see the paper [17] for more definitions and explanations).
To the memory of Olga Alexandrovna Ladyzhenskaya.
27 Page 2 of 13 G. Seregin JMFM Our standing assumption is that a suitable weak solution v and q to the Navier-Stokes equations in Q = C×] − 1, 0[ is axially symmetric with respect to x 3 -axis. It means the following: if we introduce the corresponding cylindrical coordinates (r, θ, x 3 ) and use the corresponding representation v = v r e r +v θ e θ + v 3 e 3 , then v r,θ = v θ,θ = v 3,θ = q ,θ = 0. Here, the comma in lower indices means the partial derivative in the indicated spatial direction.
In a sense, our note is a continuation of author's paper [18], which, in turn, has been inspired by paper [15] of X. Pan, where the regularity of solutions has been proved under a slightly supercritical assumption. The aims of the present paper are to consider a local setting and a different supercritical assumption.
In order to describe our supercritical assumption, additional notation is needed. Given x = (x 1 , x 2 , x 3 ) ∈ R 3 , denote x = (x 1 , x 2 , 0). Next, different types of spatial cylinders will be denoted as 10 3 dz 3 10 for any 0 < R ≤ 1, with v = v r e r + v 3 e 3 , and assume that: for all 0 < R ≤ 2/3, where c * and α are positive constants and α obeys the condition: In the paper [18], the following completely local result has been stated. Unfortunately, the above theorem has not been proven in [18]. Instead, a global version of it has been stated and demonstrated there. However, a big step toward a proof of Theorem 1.2 has been made in that paper [18]. It is a careful analysis of the scalar equation Let us recall known differentiability properties of σ. Most of them follow from the partial regularity theory developed by Caffarelli-Kohn-Nirenberg in their famous paper [1]. As to further discussions and improvements, see also papers [12] and [7] and of course references therein.
Indeed, since v and q are an axially symmetric suitable weak solution in Q, there exists a closed subset S σ of Q, whose 1D-parabolic measure in R 3 × R is equal to zero and x = 0 for any z = (x, t) ∈ S σ , such that any spatial derivative of v with respect to Cartesian coordinates (and thus of σ) is Hölder continuous in Q \ S σ in space-time.
One can show also that σ ∈ L ∞ (Q(R)) for any 0 < R < 1, see, for example, papers [19] and [17]. What actually has been proved in paper [18], see also the last section of the present paper, is as follows: Here, osc z∈Q(r) σ(z) = M r − m r and Now, the main task of the note is to deduce Theorem 1.2 from Proposition 1.3. Let us describe briefly the most important counter-parts of our arguments. The first one is a choice of a cut-off function that takes into account the well known partial regularity of axisymmetric flows. The second important point is a local regularity properties of solutionv to the following elliptic system: In Section 3, we show how Pan type results on supercritical regularity, see Pan's paper [15], can be deduced from Theorem 1.2 of the present note. To be a bit more precise, let us replace the main supercritical condition (1.2) with the other one, where, for simplicity, it is assumed that: in Q, with a positive number α. As it follows from Theorem 1.2, supercritical condition (1.7) implies regularity of the velocity field v at the origin z = 0 for sufficiently small α.
At the end of the section, we would like to comment our notation. All absolute constants are denoted by c, other constants are denoted by C with indication of variables of which those constants may depend on. The norms of the Lebesgue space L p (ω) are denoted by · p,ω and of the mixed Lebesgue space

Proof of Theorem 1.2
Step 1. Construction of a cut-off function. The partial regularity theory for the Navier-Stokes equations implies that if singular points of an axisymmetric velocity field v exist, they must belong to the axis of symmetry, which is x 3 -axis in our case. Since the 1D parabolic Hausdorff measure of the set of singular points is equal to zero, there exist at least two regular points z 1 = (x , x 3 , t) = (0, h 1 , 0) and z 2 = (0, −h 2 , 0) of v such that 0 < h 1 , h 2 < 1. According to the properties of regular points, there are cylinders Q 1 = Q(z 1 , δ) and It is also possible to pick up r 0 close to one so that r 0 ≥ max{h 1 − δ, h 2 − δ}.
Next, exploiting the properties of singular integrals, one can derive two estimates: Again, it is important to notice that f (r, x 3 , t) = 0 only if |∇η(r, x 3 , t)| > 0. In the set supp |∇η| , functions v, ∇v, and ∇ 2 v are bounded in space-time. The most dangerous term on the right hand side of the latter inequalities are those where cut-off function η does not contain derivatives in r. These terms contain v 3 , v r, 3 , v 3,r , v 3,rr , v 3,r /r, v r,3r , and v r,3 /r. All of them are bounded by either |v|, or |∇v|, or As to the function W , the global estimates have been already established in [3]. Here, they are: This completes the proof of the lemma.
Step 3. Local estimates of solutions. In this section, our goal is to make arguments of papers [24] and [3] completely local. It is easy to verify that functions Φ = ω r /r = −v θ,3 /r and Γ = ω θ /r = (v r,3 − v 3,r )/r satisfy the following equations: Let us multiply the first equation by Φη 6 and the second equation by Γη 6 . After integration by parts, we find: Now, we wish to evaluate quantities A i and B i , starting with A 1 and B 1 and letting Ψ = ∂ t η 6 + Δη 6 to simplify our notation. Indeed, by the construction of our cut-off function η, the solution v is smooth in the domain where |Ψ| > 0. Now, it remains to use inequality |Γ| 2 + |Φ| 2 ≤ |∇ω| 2 and boundedness of |∇ω| in the corresponding space-time domain. So, we have In order to estimate A 2 and B 2 , boundedness of v and its spatial derivatives over the support of ∇η and the obvious identity are used. So, we have Since |Γ| 2 + |Φ| 2 ≤ |∇ω| 2 and |∇ω| is bounded in { 1 2 < |x | < 1}, the estimate is easily derived.
Our next goal is to find bounds for A 3 and B 3 . To this end, let us fix a number 0 < r 1 < 1/4 and introduce domain S 1 = {x ∈ C : |x | < r 1 }. Then, In order to estimate the right hand side of the latter inequality, we are going to use a Leray type inequality in dimension two. For the reader convenience, let us state it for our particular case. The proof can be done with the help of integration by parts.

Lemma 2.2.
For any function f ∈ C 1 0 (C), the following inequality is valid: Applying Lemma 2.2, we find Now, let us exploit one more time the fact that our solution v has bounded spatial derivatives of any order in the support of ∇η and inequality |Γ| 2 + |Φ| 2 = (|ω r | 2 + |ω θ | 2 )/r 2 ≤ |∇ω| 2 . So, As to the last quantity A 3 , we find The first term on the right hand side can be transformed so that: Since |v r /r| ≤ |∇v| and since the second integral is taken over support of ∇η, we find To evaluate A 0 , we may use Hölder inequality and the same trick as above. It gives us to the bound: Here, it has been used boundedness of ∇ 2 v and ∇ 2 ω in domain {r > r 1 }×] − 1, 0[ Now, we again apply Lemma 2.2 and find another bound: It remains to take into account the statement of Lemma 2.1 and conclude Our next aim is a bound for A 32 . Obviously, we have Here, we would like to use again that v, ∇v, and ∇ 2 v are bounded over the support of ∇η. In addition, we know that |v θ | ≤ |v|, |Φ| ≤ ∇ω, |(v r /r) ,3 | ≤ |∇ 2 v|, and |(v r /r) ,3 | ≤ |∇ 2 v|. Therefore, we find So, finally, Combing all the estimates made on this step, we shall have: Here, it is assumed that a number r 1 ∈]0, 1/4[ so small as cC 1 ln(e/r 1 ) + cC 2 1 ln 4 (e/|r 1 |) < 2.
So, if the latter condition holds, the key estimate can be derived by more or less standard arguments. It is as follows: Step 4. Final Conclusion. Now, let us show that the origin z = 0 is a regular point of v. To this end, we are going to use the estimate proved at the previous step. It can be re-written to the form: where as usual |f | 2 2,Q = sup −1<t<0 f (·, t) 2 2,C + ∇f 2 2,Q . Since ω θ = rΓ, one can state that |η 3 ω θ | 2,Q < ∞ as well. It will be used to make various estimates of the solution to equation (2.3). Indeed, the elliptic theory implies two classical bounds: 3 2,C + c |∇η 3 ||v| 2,C and ∇ 2 (η 3 v) 2,C ≤ c |∇η 3 ||∇v| 2,C + c |∇ 2 η 3 ||v| 2,C + c curl(ω θ η 3 e θ ) 2,C . Then, our previous arguments can be exploited to describe properties of the solution v in the support of ∇η and conclude that the quantity |∇(η 3 v)| 2,Q is bounded that in turn yields boundedness of two norms: ∇(η 3 v) 10 3 ,Q and ∇(η 3 v) ∞, 10 3 ,Q . So, for all sufficiently small R, we find 1 as R → 0. It remains to understand what happens with v θ . Indeed, we have The latter inequality can be re-written so that 3 |v θ |/r) 10 3 (r, y, t)dy 3 10 .
Taking into account the inequality |v θ |/r ≤ |∇v| and boundedness of |∇v| in the support of ∇η, after integration by parts, we get: So, for sufficiently small R > 0, we have Here, we are going to use two scale-invariant inequalities proved in [20]. They are as follows: From the above inequalities, it can be easily derived Next, we choose a positive number θ to satisfy the inequality cθ ≤ 1/4 and then pick up a positive number ε so that ε θ 2 + To estimate the above integral, we assume that a number α and the variable are positive and sufficiently small. Then, integration by parts gives us: