A Slightly Supercritical Condition of Regularity of Axisymmetric Solutions to the Navier–Stokes Equations

In the note, a new regularity condition for axisymmetric solutions to the non-stationary 3D Navier–Stokes equations is proven. It is slightly supercritical.


Introduction
In this note, we continue to analyse potential singularities of axisymmetric solutions to the non-stationary Navier-Stokes equations. In the previous paper [24], it has been shown that an axially symmetric solution is smooth provided a certain scale-invariant energy quantity of the velocity field is bounded. By definition, a potential singularity with bounded scale-invariant energy quantities is called the Type I blowup. It is important to notice that the above result does not follow from the so-called ε-regularity theory developed in [2,16], and [10], where regularity is coming out due to smallness of those scale-invariant energy quantities.
This article is part of the topical collection "Yoshihiro Shibata" edited by Tohru Ozawa.
Actually, our note is inspired by the paper [20], where the regularity of solutions has been proved under a slightly supercritical assumption. We would like to consider a different supercritical assumption, to give a different proof and to get a better result.
To state our supercritical assumption, additional notation is needed. Given for any 0 < R ≤ 1 and assume that: for all 0 < R ≤ 2/3, where c * and α are positive constants and α obeys the condition: Without loss of generality, one may assume that g(R) ≥ 1 for 0 < R ≤ 2 3 . To ensure the above condition, it is enough to increase the constant c * if necessary.
Our aim could be the following completely local statement. However, in this paper, we shall prove a weaker result leaving Theorem 1.2 as a plausible conjecture. We shall return to a proof of Theorem 1.2 elsewhere. In the present paper, the following fact is going to be justified.

Theorem 1.3. Let v be an axially symmetric solution to the Cauchy problem for the Navier-Stokes equa-
and for all 0 < R ≤ 2/3 (it is assumed for simplicity that T > 1), with some positive constants c * and α, .
Then v is a strong solution to the above Cauchy problem in Our proof is based on the analysis of the following scalar equation Let us list some differentiability properties of σ. Some of them follows from partial regularity theory developed by Caffarelli-Kohn-Nirenberg.
Indeed, since v and q are an axially symmetric suitable weak solution, there exists a closed set S σ in 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 (and thus of σ) is Hölder continuous in Q \ S σ .
Next, we observe that Since v is axially symmetric, the first factor on the right hand side is finite. This fact, by iteration, yields σ ∈ W 2,1 p (P (δ, R; R)×] − R 2 , 0[) for any 0 < δ < R < 1 and for any finite p ≥ 2.
It follows from the above partial regularity theory that, for any −1 < t < 0, In the same way, as it has been done in [26] and [24], one can show that σ ∈ L ∞ (Q(R)) for any 0 < R < 1.
The main part of the proof of Theorem 1.3 is the following fact.
where c is a positive absolute constant and 0 < r < R ≤ R * (c * , α) ≤ 1/6. Here, osc z∈Q(r) σ(z) = M r − m r and The above statement is an improvement of the result in [20], where the bound for oscillations of σ contains a fixed power of logarithmic factor only. The proof of Proposition 1.4 is based on a technique developed in [18], see also references there. We also would like to mention interesting results for the heat equation with a divergence free drift, see [1,6,7,25].

Auxiliary Facts
Define the class V of functions π : Q → R possessing the properties: (i) there exists a closed set S π in Q, whose 1D-parabolic measure R 3 × R is equal to zero and x = 0 for any z = (x , x 3 , t) ∈ S π , such that any spatial derivative is Hölder continuous in Q \ S π ; (ii) We are going to use the following subclass V 0 of the class V, saying that π ∈ V 0 if and only if π ∈ V and . Then π has the property (B r ) in Q(2r) with any constant less or equal to k R .

Remark 2.3.
If we additionally assume that π(·, −θR 2 ) ≥ k in B for some 0 < θ ≤ 1, then we do not need to use a cut-off in t. So, for 0 < λ < 1, we have

Corollary 2.4. Let a non-negative function
for some t 0 > −4R 2 , for some 0 < k ≤ k R , and for some 0 < μ ≤ μ * = 1 2c 1 (λ 1 , λ, θ/2, θ, M(2R)) 10 3 . t 0 ), R). Proof. The first statement can be proved ad absurdum with the help of inequality (2.2) and a suitable choice of the number μ * . The second statement is proved in the same way but with the help of the inequality of Remark 2.3. Number μ * is defined by the constant c 1 instead of c 1 . The two lemmas below are obvious modifications of the corresponding statements in the paper [18].

Proof of Proposition 1.4
Now, we can state an analog of Lemma 4.2 of [18] for the class V.