On the Blowup for the 3D Axisymmetric Incompressible Chemotaxis-Euler Equations

Here, n and u represent the concentration of bacteria and the velocity field of the transported water, respectively. Besides, the vector field u is divergence free and independent of n. The equation describes the evolution of the bacteria transported by the velocity field of the fluid. Moreover, these cells are attracted by the oxygen concentration generated by chemotaxis. For the term χ∇·ðn∇ ðΔÞ−1cÞ, χ > 0 is a parameter controlling the influence of the chemotactic effect. In addition, κ is the strength growth rate of the population and μ is a parameter regulating death by overcrowding. Apart from Equation (1), there are a lot of other models illustrating the procedure of oxygen attraction in biology. An increasing number of mathematicians studied the process in the past years, see [7–15]. Our aim in this paper is to further explore model (1), combined with an oxygen equation and a Navier-Stokes equation, see [16]. Then, we obtain the following model in d, d = 2, 3,


Introduction
The effect of oxygen attraction on the emergence of bioconvective patterns is studied in [1,2]. Some experiments, such as a colony of Bacillus subtilis suspending in a drop of water, are carried out to identify this phenomenon. From paper [3][4][5][6], we can also find the important role of chemotaxis between sperm and eggs. The following model, in [1], is introduced to analyze the above phenomenon: Here, n and u represent the concentration of bacteria and the velocity field of the transported water, respectively. Besides, the vector field u is divergence free and independent of n. The equation describes the evolution of the bacteria transported by the velocity field of the fluid. Moreover, these cells are attracted by the oxygen concentration generated by chemotaxis. For the term χ∇·ðn∇ ðΔÞ −1 cÞ, χ > 0 is a parameter controlling the influence of the chemotactic effect. In addition, κ is the strength growth rate of the population and μ is a parameter regulating death by overcrowding.
Apart from Equation (1), there are a lot of other models illustrating the procedure of oxygen attraction in biology. An increasing number of mathematicians studied the process in the past years, see [7][8][9][10][11][12][13][14][15]. Our aim in this paper is to further explore model (1), combined with an oxygen equation and a Navier-Stokes equation, see [16]. Then, we obtain the following model in d, d = 2, 3, n t + u · ∇n = Δn − χ∇· n∇c ð Þ+ κn − μn 2 , c t + u · ∇c = Δc − nc, The unknowns are n, c, u, and P, standing for the bacteria, the oxygen, the velocity field, and the pressure of the fluid separately. The third equation of the above system contained an extra force, buoyancy, which is produced by the density and a given gravitational potential Φ. η is the dissipation coefficient. If κ = μ = 0 and η > 0, the global existence of weak solutions in 2 was shown in [8,12].
In this paper, we choose χ = κ = μ = 1 and η = 0, then (2) can be changed into the following one: The Euler equation is shown as the following form: In three dimensional space, the vorticity equation has the form But the chief difficulty is we are lacking information on the vortex-stretching term ω · ∇u. Although the global existence of classical solutions for the 3D Euler equation is an open problem, some known results are obtained under the circumstances of axisymmetric flows without swirl. That a vector field u is axisymmetric without swirl is defined as follows: where ðe r , e θ , e z Þ is the cylindrical basis of 3 and the components u r and u z do not depend on the angular variable. With this structure, vorticity takes the form and satisfies Hence, the quantity Γ ≔ ω θ /r obeys to the equation The goal of this paper is to build the blowup criterion of smooth solutions for (3) by the Fourier localization technique. Here, we follow ideas introduced in [17][18][19][20][21]. Our result reads as the following: Theorem 1. For s > 3, suppose the triple ðn 0 , c 0 , u 0 Þ ∈ H s × H s+1 × H s+1 and Φ ∈ H s+2 . Let u 0 be an axisymmetric divergence-free vector field and its vorticity satisfies ω 0 /r ∈ L 3,1 . Assume that are the smooth solutions to (3). If the condition holds true, then the solutions ðn, c, uÞ can be extended beyond T > 0.

Remark 2.
In paper [20], a regularity criterion in terms of two items is established. But in Theorem 1, we give a different criterion using the only bacteria concentration in 3. The bacteria concentration plays a more important role in this model, and the nonlinear term −∇ · ðn∇cÞ is difficult to estimate. Hence, using bacteria concentration to show the regularity is natural and physical.
Notation. Throughout the paper, C means a harmless constant and may vary from line to line; C T denotes a constant C relating to T; ∥·∥ p stands for the norm of the Lebesgue space L p .

Preliminaries
In this section, we give the definition of some function spaces and recall some useful lemmas. Firstly, we use the dynamic partition of the unity to give the definition of Besov spaces. One may check [22] for exact details. Let φ ∈ C ∞ 0 ðdÞ be set in C = fξ ∈ d, 3/4≤|ξ|≤8/3g satisfying Let χðξÞ = 1 − ∑ q∈N φð2 −q ξÞ. For f ∈ S ′ , Littlewood-Paley operators are defined as follows: The low-frequency cut-offs are denoted: Now, we introduce the definition of the Besov space. For Journal of Function Spaces defined as the set of tempered distributions of f ∈ S ′ /P satisfying where P is the polynomial space. The inhomogeneous space B s p,r is the set of tempered distribution f with the norm It is worthwhile to remark that B s 2,2 and B s ∞,∞ coincide with the usual Sobolev spaces H s and the usual Hölder space C s for s ∈ ℝ \ ℤ, respectively.
In our study, we require the space-time Besov spaces as the following manner: for T > 0 and ρ ≥ 1, we denote by L ρ T B s p,r the set of all tempered distribution f such that Lemma 3 (see [22]).
ð18Þ Lemma 4 (see [22]). There exists a constant C > 0 such that for s > 0, we have Lemma 5 (see [23]). Let u be a solution of the transport equation There exists a sequence c q ∈ ℓ r ðℤÞ such that ∥c q ∥ ℓ r = 1 and a constant C depending only on d, r, s, p, and p 1 , which satisfy with Lemma 6 (see [24]). Let ½p, r ∈ ½1,∞ 2 , v be a divergence-free vector-field belonging to the space L 1 loc ðℝ + ; Lipðℝ d Þ and let a be a smooth solution of the following transport equation: If the initial data a 0 ∈ B 0 p,r , then we have for all t ∈ +ℝ 3. Proof of Theorems 3.1. Local Well-Posedness. We construct the following smoothing system: Step 1. Uniform boundedness.
Taking the operation Δ q with q ≥ −1 on the first equation of (25), we obtain Δ q n k t + Δ q u k · ∇n k = ΔΔ q n k −∇ · Δ q n k ∇c k + Δ q n k − Δ q n k 2 :

Journal of Function Spaces
Making the L 2 -inner product for (26) with Δ q n k yields Multiplying 2 2qs on both sides of the above inequality, then taking the ℓ 1 norm, using Hölder's inequality and Young's inequality together with Lemma 4, we have Then, we conclude In a similar way to (29), we obtain Thus, we have Operating Δ q with q ≥ −1 to the third equation of (25) implies Taking the L 2 -inner product for the above equality with Multiplying 2 2qðs+1Þ on both sides of the above inequality and taking the ℓ 1 norm, we conclude Collecting (29)-(34), we have We obtain from the Gronwall inequality that Let then we obtain Step 2. Extracting sequences.
According to (38), we get Journal of Function Spaces In order to prove the convergence, we require uniform boundedness for ∂ t n k , ∂ t c k , and ∂ t u k . By the first equation of (25), we infer In a similar process, we have Since L 2 is locally compactly embedded in H −1 , we can apply the Aubin-Lions Lemma to deduce that, extracting a subsequence, the approximate solution sequence ðn k , c k , u k Þ strongly converges in L ∞ ð½0, T ; H −1 Þ to some function ðn, c, uÞ such that By the above estimates, we can easily have the limit in the approximate system (25) and ðn, c, uÞ solve (3) in the sense of distribution. Using a classical method [12], we have n ∈ Cð½ 0, T ; H s Þ, c ∈ Cð½0, T ; H s+1 Þ, and u ∈ Cð½0, T ; H s+1 Þ.
This completes the proof of Theorem 1.

Data Availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.