On blow-up criteria for a coupled chemotaxis fluid model

We consider a coupled chemotaxis fluid model and prove some blow-up criteria of the local strong solution.


Introduction
We consider the following coupled chemotaxis fluid model Here u denotes the velocity vector field of the fluid and π is the pressure scalar, p and n denote the concentration of oxygen and bacteria, respectively. ∇φ is the gravitation force. f (p) ≥ f () =  and χ(p) ≥  are two given smooth functions of p. When φ = , (.) and (.) are the well-known Navier-Stokes system. Kozono et al. [] and Kozono and Shimada [] proved the following blow-up criteria: HereḂ s p,q denotes the homogeneous Besov space. Zhang et al.
[] showed the following blow-up criterion in terms of pressure: When u = ∇φ = , (.) and (.) are the Keller-Segel model which was studied in [, ]. Very recently, Chae et al. [] showed the local well-posedness of smooth solutions to problem (.)-(.) and the following blow-up criterion: The aim of this paper is to refine (.) further; we will prove the following.
Theorem . Let the initial data (u  , n  , p  ) be given in H l × H l- × H l for l >   and n  , and n satisfies (.). We omit the details here.

Preliminary
Here we recall the definitions and some properties of spaces.
Choose two nonnegative smooth radial functions χ , ϕ supported, respectively, in B and C such that where F - stands for the inverse Fourier transform. Then the dyadic blocks j and S j can be defined as follows: Formally, j = S j -S j- is a frequency projection to annulus {ξ : C   j ≤ |ξ | ≤ C   j }, and S j is a frequency projection to the ball {ξ : |ξ | ≤ C j }. One can easily verify that, with our choice of ϕ, With the introduction of j and S j , let us recall the definition of the Besov space.
and S denotes the dual space of S = {f ∈ S(R d ); ∂ αf () = ; ∀α ∈ N d multi-index} and can be identified by the quotient space of S /P with the polynomials space P.

Proof of Theorem 1.1
This section is devoted to the proof of Theorem .. Since local existence results have been proved in [], we only need to prove a priori estimates.
To begin with, it is easy to see that Case . Let (.) and (.) hold true. Testing (.) by u and using (.), we infer that which leads to In the following calculations, we will use the following elegant inequality [, ]: Testing (.) by u, using (.) and the above inequality, we find that which yields (.); this completes the proof of Case  again by (.). Case . Let (.) and (.) hold true. Testing (.) byu, using (.), we deduce that By the very same calculations as those in [], we get Inserting (.) into (.) and solving the resulting inequality, we arrive at (.). This completes the proof of Case . Case . Let (.) (r = -) and (.) hold true. Testing (.) by |u|  u and using (.), we observe that I  can be bounded as follows: We bounded I  as follows: where we have used the elegant inequality [, ] and the pressure estimate Inserting (.) and (.) into (.) and using the Gronwall inequality, we conclude that Here we used the Gagliardo-Nirenberg inequality Now, since the proofs of other cases are very similar to those in Case , Case , Case  and Case , we only prove the following case: Let (.) (- < r ≤ ) and (.) hold true.