Abstract

In this paper, we investigate the 3D incompressible chemotaxis-Euler equations. Taking advantage of the structure of axisymmetric fluids, we establish the blowup criterion of the system using the Fourier localization method.

1. 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 [3–6], 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, and represent the concentration of bacteria and the velocity field of the transported water, respectively. Besides, the vector field is divergence free and independent of . 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 , is a parameter controlling the influence of the chemotactic effect. In addition, is the strength growth rate of the population and 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 , ,

The unknowns are , , , and , 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 . is the dissipation coefficient. If and , the global existence of weak solutions in was shown in [8, 12].

In this paper, we choose and , 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 . 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 is axisymmetric without swirl is defined as follows: where is the cylindrical basis of and the components and do not depend on the angular variable. With this structure, vorticity takes the form and satisfies

Hence, the quantity 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 [17–21]. Our result reads as the following:

Theorem 1. For , suppose the triple and . Let be an axisymmetric divergence-free vector field and its vorticity satisfies . Assume that are the smooth solutions to (3). If the condition holds true, then the solutions can be extended beyond .

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 . The bacteria concentration plays a more important role in this model, and the nonlinear term is difficult to estimate. Hence, using bacteria concentration to show the regularity is natural and physical.

Notation. Throughout the paper, means a harmless constant and may vary from line to line; denotes a constant relating to ; stands for the norm of the Lebesgue space .

2. 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 be set in satisfying

Let . For , Littlewood-Paley operators are defined as follows:

The low-frequency cut-offs are denoted:

Now, we introduce the definition of the Besov space. For , the homogenous Besov space is defined as the set of tempered distributions of satisfying where is the polynomial space. The inhomogeneous space is the set of tempered distribution with the norm

It is worthwhile to remark that and coincide with the usual Sobolev spaces and the usual Hölder space for , respectively.

In our study, we require the space-time Besov spaces as the following manner: for and , we denote by the set of all tempered distribution such that

Lemma 3 (see [22]). Let . Suppose that , then there exists a constant independent of , such that

Lemma 4 (see [22]). There exists a constant such that for , we have

Lemma 5 (see [23]). Let be a solution of the transport equation and define , , , and There exists a sequence such that and a constant depending only on , and , which satisfy with

Lemma 6 (see [24]). Let , be a divergence-free vector-field belonging to the space and let be a smooth solution of the following transport equation: If the initial data , then we have for all

3. Proof of Theorems

3.1. Local Well-Posedness

We construct the following smoothing system:

Step 1. Uniform boundedness.

Taking the operation with on the first equation of (25), we obtain

Making the -inner product for (26) with yields

Multiplying on both sides of the above inequality, then taking the 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 with to the third equation of (25) implies

Taking the -inner product for the above equality with gives

Multiplying on both sides of the above inequality and taking the 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