Global solutions to the Chemotaxis--Navier--Stokes equations with some large initial data

In this paper, we mainly study the Cauchy problem of the Chemo-taxis-Navier-Stokes equations with initial data in critical Besov spaces. We first get the local wellposedness of the system in \begin{document}$\mathbb{R}^d \, (d≥2)$\end{document} by the Picard theorem, and then extend the local solutions to be global under the only smallness assumptions on \begin{document}$\|u_0^h\|_{\dot{B}_{p, 1}^{-1+\frac{d}{p}}}$\end{document} , \begin{document}$\|n_0\|_{\dot{B}_{q, 1}^{-2+\frac{d}{q}}}$\end{document} and \begin{document}$\|c_0\|_{\dot{B}_{r, 1}^{\frac{d}{r}}}$\end{document} . This obtained result implies the global wellposedness of the equations with large initial vertical velocity component. Moreover, by fully using the global wellposedness of the classical 2D Navier-Stokes equations and the weighted Chemin-Lerner space, we can also extend the obtained local solutions to be global in \begin{document}$\mathbb{R}^2$\end{document} provided the initial cell density \begin{document}$n_0$\end{document} and the initial chemical concentration \begin{document}$c_0$\end{document} are doubly exponential small compared with the initial velocity field \begin{document}$u_0$\end{document} .


(Communicated by Pierre Germain)
Abstract. In this paper, we mainly study the Cauchy problem of the Chemotaxis-Navier-Stokes equations with initial data in critical Besov spaces. We first get the local wellposedness of the system in R d (d ≥ 2) by the Picard theorem, and then extend the local solutions to be global under the only smallness assumptions on u h . This obtained result implies the global wellposedness of the equations with large initial vertical velocity component. Moreover, by fully using the global wellposedness of the classical 2D Navier-Stokes equations and the weighted Chemin-Lerner space, we can also extend the obtained local solutions to be global in R 2 provided the initial cell density n 0 and the initial chemical concentration c 0 are doubly exponential small compared with the initial velocity field u 0 .
1. Introduction. In the paper, we consider the following Chemotaxis-Navier-Stokes equations in R d (d ≥ 2): n t − γ∆n + u · ∇n = −∇ · (χ(c)n∇c), (1.4) with the initial conditions where u(x, t) : R d × R + → R d is the fluid velocity field, n = n(x, t) : R d × R + → R + is the cell density, c = c(x, t) : R d × R + → R + is the chemical concentration, and P = P (x, t) : R d × R + → R is the pressure of the fluid. Positive constants γ, µ and ν are the corresponding diffusion coefficients for the cells, chemical and fluid, respectively. χ(c) is the chemotactic sensitivity and κ(c) is the consumption rate of the chemical by the cells. φ = φ(x) is a given potential function accounting the effects of external forces such as gravity. Different functional forms of χ(c) and κ(c) are meaningful, according to various conceivable threshold effects and saturation mechanism. In general, χ(c), κ(c) and φ(x) are supposed to be sufficiently smooth given functions. In the present paper, we consider a simplified model with γ = µ = ν = 1, χ(c) = 1 and κ(c) ≤ Ac for some constant A. Throughout the paper, we write u(x, t) = (u 1 (x, t), u 2 (x, t), · · ·, u d−1 (x, t), u d (x, t)) def = (u h (x, t), u d (x, t)), and call u h the horizontal component of u, and u d the vertical component of u.
Here "horizontal" and "vertical" should be understood as a convention.
The system (1.1)-(1.4) describes a biological process, in which cells (e.g. bacteria) move towards a chemically more favorable environment. Specifically, the mechanism is a chemotactic movement of bacteria often towards higher concentration of oxygen which they consume, a gravitational effect on the motion of the fluid by the heavier bacteria, and a convective transport of both cells and oxygen through the water. One can refer to [14], [22] for more details. This model has been studied by many researchers due to the significance of the biological background. For the case χ(c) = c, Lorz [14] got the local existence of the solutions for the Keller-Segel-Stokes system in R 2 and R 3 . Duan, Lorz, and Markowich [8] obtained the global-in-time existence of the H 3 (R d )-solutions, near constant states, to (1.1)-(1.4), i.e., if the initial data (n 0 −n ∞ , c 0 , u 0 ) H 3 is sufficiently small, then there exists a unique global solution. Moreover, they showed the global existence of the weak solutions to the Keller-Segel-Stokes equations in R 2 , where the nonlinear convective term u · ∇u is ignored in (1.3), with the small data assumptions on either ∇φ or the initial data c. The key ingredient of their proof is to establish a priori estimates involving energy type functionals. Subsequently, under the same assumptions on χ and κ [8], Liu and Lorz [13] removed the above smallness conditions and proved the global existence of weak solutions to the 2D Keller-Segel-Navier-Stokes equations for arbitrarily large initial data. In [23], Winkler proved that the system (1.1)-(1.4) admits a unique global classical solution in a bounded convex domain with smooth boundary in R 2 under the assumptions on χ, κ, φ which are weaker than those in [8], [13]. Chae, Kang and Lee [6] established the local existence of regular solutions and presented some blow-up criteria in H m (R d ) (d = 2, 3) with m ≥ 3. They also got the existence of global weak solutions in R 3 with stronger restrictions on the consumption rate and the chemotactic sensitivity. However, whether the solutions to (1.1)-(1.4) with large initial data exist globally or blow up in finite time remains an open problem.
The main goal of this paper is to show the local and global wellposedness of (1.1)-(1.4) with initial data in critical Besov spaces. Here "the critical space" means that we want to solve the system (1.1)- (1.4) in functional spaces with invariant norms by the changes of scales that leave (1.1)-(1.4) invariant. For (1.1)-(1.4), it is easy to see that the transformations (u λ , n λ , c λ )(t, x) = (λu(λ 2 ·, λ·), λ 2 n(λ 2 ·, λ·), c(λ 2 ·, λ·)) have this "critical" property provided that the pressure term P and the potential function φ(x) have been changed accordingly. We can also verify that the product r,1 (R d ), 1 ≤ p, q, r ≤ ∞, is the critical space for the system (1.1)-(1.4). By fully using Bony's decomposition, Bernstein's lemma and interpolation inequality in the Chemin-Lerner space, we get respectively different estimates of couple terms appeared in (1.1)-(1.4). Then we can establish the local wellposedness of (1.1)-(1.4) by the Picard theorem in R d (d ≥ 2). Moreover, using the incompressible condition divu = 0, anisotropic Bernstein's lemma and Gagliardo-Nirenberg's inequality, we can extend our local solutions to be global under the only smallness assumptions on u h . It should be mentioned that, our global wellposdness results do not require any size restrictions on the third component of the initial velocity u d 0 . Our first main result in this paper is stated as follows: is small enough, and there exists a positive constant for 0 < ε < 2d p − 1, then the local solution can be extended to be global.
Remark 1. Since the system is isotropic, the "vertical" speed u d can be regarded as any one of the components of u.
Remark 2. The main reason we need to impose on the restrictive condition p, q, r list in Theorem 1.1 lies in the fact that we need to use Bony's paraproduct decomposition to deal with the coupled terms in (1.1)-(1.4).
As a by-product of Theorem 1.1, we can give a blow-up criterion.
Let ω = ∇ × u and let T * be the maximal local existence time of (u, n, c) in Theorem 1.1. If T * < ∞, then We now present our second main result of the paper. More precisely, we can obtain the following global wellposedness result without any size restrictions on the initial velocity u 0 in R 2 .
is small enough. If there exist positive constants C 0 , ε 0 such that the initial data (u 0 , n 0 , c 0 ) satisfies (1.10) The remainder of the paper is organized as follows. In Section 2, we recall some basic facts on the Littlewood-Paley theory and various product laws in Besov spaces. In Section 3, we present the proof of the local wellposedness result in Theorem 1. In Section 4, by anisotropic Bernstein's lemma and Gagliardo-Nirenberg's inequality, we can extend our local solutions to be global under the some smallness condition on initial data. In Section 5, we give the proof of Corollary 1.4. In the last section, by fully using the global wellposedness of the classical 2D Navier-Stokes system and the weighted Chemin-Lerner space, we also get the global wellposedness of (1.1)-(1.4) in R 2 under another type of initial data. Notations. Given A, B are two operators, we denote the commutator [A, B] = AB − BA. For a b, we mean that there is a uniform constant C, which may be different on different lines, such that a ≤ Cb. For a Banach space X and an interval I of R, we denote by C(I; X) the set of continuous functions on I with values in X. For q ∈ [1, +∞], the notation L q (I; X) stands for the set of measurable functions on I with values in X, such that t → f (t) X belongs to L q (I). Finally, we let (d j ) j∈Z denote a generic element of 1 (Z) such that j∈Z d j = 1.
2. Littlewood-Paley theory. In this section, we recall some basic facts on Littlewood-Paley theory (see [1] for instance). Let χ, ϕ be two smooth radial functions valued in the interval [0,1], the support of χ be the ball B = {ξ ∈ R d : |ξ| ≤ 4 3 } while the support of ϕ be the annulus C = {ξ ∈ R d : 3 4 ≤ |ξ| ≤ 8 3 }, and The homogeneous dyadic blocks∆ j and the homogeneous low-frequency cutoff operatorsṠ j are defined for all j ∈ Z bẏ Denote by S h (R d ) the space of tempered distributions u such that lim j→−∞Ṡ j u = 0 in S .

GLOBAL SOLUTIONS TO THE CNS EQUATIONS 2833
Then we have the formal decomposition Moreover, the Littlewood-Paley decomposition satisfies the property of almost orthogonality: Remark 3. Let s ∈ R, 1 ≤ p, r ≤ ∞ and u ∈ S h (R d ). Then there exists a positive constant C such that u belongs toḂ s p,r (R d ) if and only if there exists {c j,r } j∈Z such that c j,r ≥ 0, c j,r r = 1 and ∆ j u L p ≤ Cc j,r 2 −js u Ḃs p,r , ∀j ∈ Z.
If r = 1, we denote by d j def = c j,1 .
Definition 2.2. Let s ∈ R and 0 < T ≤ +∞. We define , and with the standard modification for r = ∞ or σ = ∞.
The following anisotropic Bernstein's lemma will be repeatedly used throughout this paper.
If the support of f is included in 2 k B h , then If the support of f is included in 2 B v , then If the support of f is included in 2 k C h , then If the support of f is included in 2 C v , then . On the other hand, it has been demonstrated that the Bony's decomposition [1,2] is very effective to deal with nonlinear problems. Here, we recall the Bony's decomposition in the homogeneous context: We shall also use the following lemmas to prove our theorems. (2.12) Proof. This lemma is proved in [17] for the case when q ≤ p. We shall only prove (2.12) for the case when q > p. Applying Bony's decomposition, we have Then by Lemma 2.4, we get for , and for This completes the proof of the lemma.
Then there holds and . (2.14) Proof. The proof of this lemma is similar to that of Lemma 2.4 in [27], we also outline its proof here for completeness. Firstly, using divu = 0 and the Gagliardo-Nirenberg inequality, we have . Similarly, .
This completes the proof of the lemma.

(2.23)
To estimate the remaining termṘ(v d , div h w h ), we consider the following two cases. .
Then for any X 0 ∈ O, there exists a time T such that the ODE 3. Local wellposedness of Theorem 1.1. Now we are in a position to prove Theorem 1.1. Let p, q, r satisfy the conditions in Theorem 1.1 with divu 0 = 0, Then we introduce a vector space X T def = X T × Y T × Z T with the following product norm: (3.26) respectively equipped with the norms .
We shall denote the heat semiflow by S(t) = e t∆ , and the projector by P = Id + (−∆) −1 ∇ div . Let Ψ : X T → X T be a map given by LetB δ be a closed ball centered at 0 in X T and the radius δ will be specified later. Hence, the proof of Theorem 1.1 is reformulated to prove that the map Ψ is a contraction mapping inB δ for some δ. By Lemma 2.9, for any 0 < t ≤ T , ( 3.27) In what follows, we will present three lemmas to deal with the couple terms in (1.1)-(1.4).
Proof. By Lemma 2.5, one can get easily .

XIAOPING ZHAI AND ZHAOYANG YIN
Again by Lemma 2.5 and interpolation inequality in the Chemin-Lerner space, we have This complete the proof of the lemma. .
This complete the proof of the lemma. ).
Proof. Similar to the proof of the above lemma, one can infer from Lemma 2.5 that . (3.28) When r ≥ p, one can deduce from Lemma 2.5, Remark 4 and Young's inequality that Thus, in the following we only pay attention to the case r ≤ p. According to Bony's decomposition, we have u · ∇c =Ṫ u ∇c +Ṫ ∇c u +Ṙ(u, ∇c).
This completes the proof Lemma 3.3. Now, we continue to prove the local wellposedness of Theorem 1.1. By Lemma 3.1 and the fact that P is a multiplier operator of order of zero, one has Similarly, by Lemma 3.2, one has Hence, if (u, n, c) ∈B δ and ∇φ ∈Ḃ Moreover, if we choose T small enough and we can deduce that Ψ((u, n, c)) X T δ.
In what follows, we only need to prove that the map Ψ is a contraction mapping in B δ for some δ. For that purpose, let (u 1 , n 1 , c 1 ), (u 2 , n 2 , c 2 ) be inside the ballB δ .
By Lemmas 2.5, 3.1, 3.2, 3.3, we can easily get (3.34) By the estimate (3.27) and the dominated convergence theorem, we can deduce, as T goes to 0, that S(t)(u 0 , n 0 , c 0 ) X T tends to 0. Thus we get from (3.34) that which implies that we can choose δ and T small enough such that the map Ψ is a contraction mapping inB δ . By Lemma 2.10, we can deduce that the system (1.1)-(1.4) has a unique local solution. By a standard unique continuation argument, we can obtain that 4. Global wellposedness of Theorem 1.1. The goal of this section is to prove global wellposedness of Theorem 1.1. We will divide the proof into six subsections. We first use the equation (1.3) to estimate the pressure function which will be used in the estimates of u h , u d in second and third subsections. In Subsections 4 and 5, we apply our key Lemma 2.7 to estimate u, c respectively. Finally, we complete the proof of the global wellposedness of Theorem 1.1 in Subsection 6. The main idea used in this section comes from papers [11], [17], [27]. As the authors pointed out in [17], they mainly used the algebraical structure of the momentum equation, i.e., the equation on the vertical component of the velocity is a linear equation with coefficients depending on the horizontal components. Therefore, the equation on the vertical component does not demand any smallness condition. While the equation on the horizontal component contains bilinear terms in the horizontal components and also terms associated with the interactions between the horizontal components and the vertical one. In order to solve this equation, we need a smallness condition on n, c and the horizontal component (amplified by the vertical component) of the initial data.

4.1.
The estimate of the pressure. It is well known, the main difficulty in the study of the well-posedness of incompressible system is to derive the estimate for the pressure term. We first get by taking div to the equation (1.3) and using divu = 0 that where, for a vector field u = (u h , u d ), div h u h = ∂ 1 u 1 + ∂ 2 u 2 + · · · + ∂ d−1 u d−1 . Next, we will give the estimate of the pressure which will be used in the estimates of u h and u d .
. (4.37) Proof. As both the existence and uniqueness parts of Proposition 4.1 basically follow from the uniform estimate (4.37) for appropriate approximate solutions of (4.36) . For simplicity, we just prove (4.37) for smooth enough solutions of (4.36). Firstly, according to (4.36), we have Applying the operator∆ j to the above equation (4.38), taking L 1 t (L p )-norm and using Lemma 2.4, we obtain ∆ j (∇Π) L 1 ), (4.45) Applying∆ j to (4.45) and taking L 2 inner product with |∆ j u h | p−2∆ j u h (when 1 < p < 2, we need to make some modification, see [7]), we obtain where we have used the following fact (it can be found in the Appendix of [7]): there exists a positive constant C 1 so that By using divu = 0, we get Thus, by Lemmas 2.4, 2.5, 2.8 and the estimates (4.37), (4.41)-(4.43) , we have ). (4.47) Hence, integrating (4.46) over [0, t] and taking the above estimate into the resulting inequality, we can infer that .
Applying the operator∆ j to the above equation and taking L 2 inner product of the resulting equation with |∆ j u d | p−2∆ j u d , respectively, we get by a similar derivation of (4.46) that (4.50) By using divu = 0, (4.43) and Lemma 2.4, we have ). (4.51) Hence, integrating (4.50) over [0, t] and taking estimates (4.37), (4.51) into the resulting inequality, we can deduce that .
(4.60) 4.5. The estimate of c. In this subsection, we will give the estimates of c. Apply-ing∆ j to (1.2) and taking L 2 inner product with |∆ j c| r−2∆ j c, we get by a similar derivation of (4.53) that Integrating the above estimate from 0 to t and using a standard commutator's argument and the basic energy estimate, one has (4.62) On the one hand, we get by applying the classical estimate on commutators and (2.14) with m 1 = m 2 = ∞ that ). (4.63) Similar to the estimate (4.58), we will divide the proof into two cases to estimate the rest termṘ(u, ∇c). When r ≥ p, thanks to Lemma 2.4 and (2.14) with m 1 = p, m 2 = ∞, we obtain ). (4.64) When r ≤ p and 1 r − 1 p ≤ 1 d , by Lemma 2.4 and (2.14) with m 1 = p, m 2 = ∞, one has ). (4.65) On the other hand, by Lemma 3.3, one has .
In what follows, we will prove that T * * = T * = ∞ under the assumption of (1.6).
If not, we assume that T * * < T * . For ∀t ≤ T * * , we get from (4.52) that .

(4.75)
While taking The goal is to show that if assumption (5.78) holds, there is a bound C depending only on u 0 , n 0 , c 0 , T * and M 0 such that and there holds (6.89) with λ ≥ 0. Recalling that the Leray projection operator to the divergence vector field space is we can deduce from the equation (6.88) that ∂ tū − ∆ū + P[ū · ∇ū +ū · ∇u R + u R · ∇ū + n∇φ] = 0.
Plugging the inequalities (6.101)-(6.103) into (6.100), we can finally get We let T * to be the maximal existence time so that (6.108)-(6.110) hold. Hence to prove Theorem 1.3, we only need to prove that T * = ∞. To complete the proof, we shall use the method of continuity. For this, we define r,1 ) ≤ η .

(6.111)
In what follows, we will prove that T * * = T * under the assumptions of (1.9).