Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces

In this article, we consider the Cauchy problem to chemotaxis model coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish the global-in-time existence of the solution when the gravitational potential ϕ and the small initial data \begin{document}$(u_{0}, n_{0}, c_{0})$\end{document} in critical Besov spaces under certain conditions. Moreover, we prove that there exist two positive constants σ0 and \begin{document}$C_{0}$\end{document} such that if the gravitational potential \begin{document}$\phi \in \dot B_{p,1}^{3/p}({\mathbb{R}^3})$\end{document} and the initial data \begin{document}$(u_{0}, n_{0}, c_{0}): = (u_{0}^{1}, u_{0}^{2}, u_{0}^{3}, n_{0}, c_{0}): = (u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$\end{document} satisfies \begin{document}$\begin{equation*}\begin{aligned} &\left(\left\|u_{0}^{h}\right\|_{\dot{B}^{-1+3/p}_{p, 1}(\mathbb{R}^3)}+\left\|\left(n_{0}, c_{0}\right)\right\|_{\dot{B}^{-2+3/q}_{q, 1}(\mathbb{R}^3) \times \dot{B}^{3/q}_{q, 1}(\mathbb{R}^3)}\right)\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\exp\left\{C_{0}\left(\left\|u_{0}^{3}\right\|_{\dot{B}^{-1+3/p}_{p, 1}(\mathbb{R}^3)}+1\right)^{2}\right\} \leq \sigma_{0}\end{aligned}\end{equation*}$ \end{document} for some \begin{document}$p, q$\end{document} with \begin{document}$1 \frac{2}{3}$\end{document} and \begin{document}$\frac{1}{\min\{p, q\}}-\frac{1}{\max\{p, q\}} \le \frac{1}{3}$\end{document} , then the global existence results can be extended to the global solutions without any small conditions imposed on the third component of the initial velocity field \begin{document}$u_{0}^{3}$\end{document} in critical Besov spaces with the aid of continuity argument. Our initial data class is larger than that of some known results. Our results are completely new even for three-dimensional chemotaxis-Navier-Stokes system.


(Communicated by Doron Levy)
Abstract. In this article, we consider the Cauchy problem to chemotaxis model coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish the global-in-time existence of the solution when the gravitational potential φ and the small initial data (u 0 , n 0 , c 0 ) in critical Besov spaces under certain conditions. Moreover, we prove that there exist two positive constants σ 0 and C 0 such that if the gravitational potential φ ∈Ḃ for some p, q with 1 < p, q < 6, 1 p + 1 q > 2 3 and 1 min{p,q} − 1 max{p,q} ≤ 1 3 , then the global existence results can be extended to the global solutions without any small conditions imposed on the third component of the initial velocity field u 3 0 in critical Besov spaces with the aid of continuity argument. Our initial data class is larger than that of some known results. Our results are completely new even for three-dimensional chemotaxis-Navier-Stokes system.
The system (1) describes a biological process, in which the swimming bacteria move towards higher concentration of oxygen according to mechanism of chemotaxis and meanwhile the movement of fluid is under the influence of gravitational force generated by bacteria themselves. Both the oxygen concentration and bacteria density are transported by the fluid and diffuse through the fluid.
The model has been extensively studied by many authors and the main issue of investigation is the existence of (1). Duan, Lorz and Markowich [12] constructed global existence of weak solutions to the Cauchy problem in spatial dimension two. For the same Cauchy problem in R 2 , Liu and Lorz [31] removed the smallness assumption and obtained global existence of weak solutions with large data when κ = 1. In bounded convex domains Ω ⊂ R 2 , Winkler [53] proved that the initialboundary value problem of (1) possesses unique global classical solutions.
Minsuk, Lkhagvasuren, Choe [35] and Lkhagvasuren, Choe [7] established the existence of solutions to (1) in critical Besov spaces. Lorz [33] studied local-in-time weak solutions in a bounded domain in R d , d = 2, 3 with no-fux boundary condition and in R 2 with inhomogeneous Dirichlet conditions for oxygen. Chae, Kang and Lee [1] established the local regular existence and presented some blow-up criterions of solutions when (u 0 , n 0 , c 0 ) ∈ H m (R d ) × H m−1 (R d ) × H m (R d ) with m ≥ 3 to the Cauchy problem of (1) in R d for d = 2, 3, in particular, for two dimensional Chemotaxis-Navier-Stokes equations, regular solutions constructed locally in time are, in reality, extended globally under some assumptions. Moreover, Lorz [34] obtained global existence of a system of the elliptic-parabolic Keller-Segel equations coupled with Stokes equations (κ = 0) with small initial data in R d (d = 2; 3). Chae, Kang and Lee [2] established the local existence of regular solutions for both cases that equations of oxygen concentration is of parabolic or hyperbolic type and also proved global existence under the some smallness conditions about initial data. Zhang [61] obtained existence and uniqueness of smooth solutions in inhomogeneous Besov spaces for (1) in R d (d = 2; 3). Zhang and Zheng [64] showed that there exist global weak solutions to the Cauchy problem of (1) in R 2 with a large class of initial data. If u = 0 in the system (1), Tao [42] showed that there exist unique, global and bounded solutions if χ is sufficiently small. For stability and asymptotic behaviors on the system (1), we refer the reader to [2,3,12,30,50,63].
For the three-dimensional case, Duan, Lorz and Markowich [12] proved the globalin-time existence of smooth solutions of (1) when the initial data is close to the constant equilibriumstates in H 3 (R 3 ). Global weak solutions of the system (1) were constructed when κ = 0 [53] or when κ = 1 [54].
For the existence results of the Chemotaxis-Navier-Stokes system with nonlinear diffusion for the cell density (a porous medium type ∆n m , instead of ∆n), see [8,13,15,33,43,44,62], and the results of which have recently been (partially) extended, including information on large time behavior and in presence of general sensitivities, by Winkler [51].
A classical model about Chemotaxis equations introduced by Patlak [40] and Keller-Segel [22,23], is given as where χ is the sensitivity and τ −1/2 represents the activation length. The system (2) has been extensively studied by many authors for the past decades; see [18,19,36,37,49]. For instance, Winkler [52] studied the finite-time blow-up for the Neumann initial-boundary value problem for the fully parabolic Keller-Segel system in bounded domains.
The numerical experiments in [34] indicate an effect of the fluid interaction on possible occurrence of blow-up in two-dimensional cases. Also, three papers by Kiselev and his collaborators [9,24,25] indicate the possibility of blow-up prevention, as well as some subtle further qualitative effects, that fluid interaction may have on cross-diffusive dynamics. Variant of (2) coupled to (Navier-)Stokes equations and including logistic sources has been studied by Espejo and his collaborators [14], Tao and Winkler [44,45]. It is clear from (1) that the coupling of chemotaxis and fluid is realized through both the transport of cells and chemical substrates u∇n, u · ∇c and the external force −n∇φ exerted on the fluid by cells. In particular, if the chemotaxis effects are ignored (n = c = 0), for Ω = R 3 and κ = 1, then the induced system (1) becomes the problems related to the classical Navier-Stokes equations: which has been widely studied during the past eighty more years. Leray [26] proved the global existence of weak solutionsof the system (3), but the uniqueness and regularity of weak solutions are remaining big open problems. It has been proved that the Cauchy problem to (3) is globally well-posed for the small initial data in a series of function spaces including particularly the following critical spaces: see Fujita and Kato [16], Kato [21], Kozono and Yamazaki [29], Koch and Tataru [27].
Before going further, we recall the functional spaces we are going to use. Let S(R 3 ) be the Schwartz class of rapidly decreasing functions, S (R 3 ) be the space of tempered distributions. Here F and F −1 denote Fourier and inverse Fourier transforms of L 1 (R 3 ) functions, respectively, which are defined by More generally, the Fourier transform of any f ∈ S (R 3 ), given by (Ff, g) = (f, Fg) for any g ∈ S(R 3 ). Let C be the annulus {ξ ∈ R 3 : 3 4 ≤ |ξ| ≤ 8 3 } and D(Ω) be a space of smooth compactly supported functions on the domain Ω. There exist radial functions χ and ϕ, valued in the interval [0, 1], belonging respectively to D(B(0, 4 3 )) and D(C), such that Define the setC = B(0, 2 3 ) + C. We have From now on, we write h = F −1 ϕ andh = F −1 χ. The homogeneous dyadic blocks ∆ j and S j are defined by Here D := (D 1 , D 2 , D 3 ) and D j : It is well-known that if either s < 3/p or s = 3/p and r = 1, thenḂ s p,r (R 3 ) is a Banach space.
Let us now recall the definition of the Chemin-Lerner space L ρ (0, T ;Ḃ s p,r (R 3 )) with 0 < T ≤ ∞, s ∈ R and 1 ≤ p, r, ρ ≤ ∞ (with the usual convention if r = ∞ or ρ = ∞). The Chemin-Lerner space is defined by with the usual modification made when r = ∞ or ρ = ∞; we equip the space and the usual modifications are needed when r = ∞ or ρ = ∞. From Minkowski's inequality, it is easy to deduce that when r ≤ ρ.
If s 1 and s 2 are real numbers such that s 1 < s 2 and θ ∈ (0, 1), then we have, for any (p, r, ρ, ρ 1 , ρ 2 ) ∈ [1, ∞] 5 and 1/ρ = θ/ρ 1 + (1 − θ)/ρ 2 , where C is a positive constant independent of f . ForḂ , we deduce that whenever 1 ≤ p ≤ ∞, the product of two functions inḂ The homogeneous paraproduct of v and u is defined by The homogeneous remainder of v and u is defined by For two tempered distributions f and g, Then, up to finitely many terms, we obtain We have the following Bony decomposition: Let . We introduce a vector space Θ T := X T × Y T × Z T and with the usual product norm and For the notational simplification, when T = ∞, we denote X ∞ := X, Y ∞ := Y, Z ∞ := Z, Θ ∞ := Θ and Θ C ∞ := Θ C . Now, we state the first results of this paper.
for some sufficiently small 0 , then the system (4) admits a unique global solution such that (u, n, c) ∈ Θ ∩ Θ C .
Remark. Recently, when 1 ≤ p < 3, Chae and Lkhagvasuren [7] showed that if φ ∈Ḃ 3/p p,1 (R 3 ) and with small norm for 1 ≤ p < 6, then system (4) exists a global-in-time solution, which in particular implies the global well-posedness result in [7]. Therefore, our initial data class is larger than that of [7] and Theorem 1.1 may be regarded as a new global existence theorem on the system (4).
We can relax the smallness condition in Theorem 1.1 so that system (4) has a unique global solution. The main object of this paper is to prove the following theorem.
There exist two positive constants σ 0 and C 0 such that if the initial data (u 0 , n 0 , c 0 ) satisfies then the system (4) admits a unique global-in-time solution such that (u, n, c) ∈ Θ ∩ Θ C . Moreover, there exist C > 0 and c 2 > 0, such that Remarks.
(i) Our results do not impose any smallest conditions on the third component u 3 0 of the initial velocity field. We emphasize that our proof uses in a fundamental way the algebraical structure of (4), that is, ∇ · u = 0, which will be one of the key ingredients in the proof of Theorem 1.2 in section 4 below. We also remark that the first equation of (4) on the vertical component of the velocity is a linear equation with coeffcients depending on the horizontal components of the velocity, n and φ. Therefore, the equation on the vertical component does not demand any smallness condition. Instead of (2), the considered problem (4) here is more related to the corresponding fluid-free variant in which, as in (4) but unlike in (2), the signal is consumed rather than produced by the cells. For that problem, in the three-dimensional case up to now also smallness conditions on the data seem to be required in order to achieve global classical solvability, which can be found in a 2012 paper by Tao [46]. According to the theories of heat equation, the present analysis of us here can not perhaps yield even less restrivtive conditions for global existence the system (4) without fluid coupling and we still need the smallness condition for initial data. However, the question whether the solutions of the system (4) with large initial data exist globally or may blow up appears to remain an open and challenging topic in the three-dimensional case.
Notations. Throughout the paper, C stands for a harmless constant, and we sometimes use the notation a b as an equivalent of a ≤ Cb. Let A and B be two operators, we denote [A; B] = AB − BA. For Banach space X and interval I, we denote by C(I; X) in the set of continuous functions on I with value in X. The symbol (d j ) j∈Z is a generic element of 1 (Z) so that d j ≥ 0 and j∈Z d j = 1.

2.
Preliminaries. The proofs of Theorem 1.1 in section 3 and likewise, Theorem 1.2 in section 4, require a lot of elementary inequalities which are summarized in the following. Lemma 2.1. [17] Let C be an annulus and B be a ball in R 3 . There exists a positive constant C such that for any nonnegative integer k, any couple (p, q) ∈ [1, ∞] 2 with q ≥ p ≥ 1, and any function u of L p (R 3 ), Lemma 2.2. [17] Let C be an annulus in R 3 . There exist two positive constants c andC such that for any p ∈ [1, ∞] and any couple (t, λ) of positive numbers, if suppf ⊂ λC, then where e t∆ is the heat operator with kernel for all x ∈ R 3 and t ∈ (0, ∞).
where C is a positive constant independent of a and b.
, where C is a positive constant independent ofā andb.
Arguing by interpolation (6) and the algebar property (7) of Besov spaces yield where C is a positive constant independent of u.
where C is a positive constant independent of n and φ.
, where C is a positive constant independent of n and c.
Assume that u ∈ X T and n ∈ Y T . Then we have where C is a positive constant independent of n and u. In particular, when r 1 = ∞, we have Proof. We first get by applying the Bony paraproduct decomposition (9) that ∆ j (u∇n) = ∆ j (T u ∇n + R(u, ∇n) + T ∇n u) . (6) and Lemma 2.1 give rise to (6) and Lemma 2.1, we get that If 1 ≤ p < q < ∞, thanks to r 1 ≥ 2, Lemma 2.1 and (6), then we arrive at To estimate the remaining term R(u, ∇n), we consider two cases: 1 p + 1 q > 1 and We can find 1 < q ≤ ∞ such that 1 q + 1 q = 1, applying Lemma 2.1 and (6), for fixed constant N 0 = 2, we infer Combining (18)- (22) and taking summation for j ∈ Z, we arrive at (17). We thus conclude the proof of Lemma 2.9.
Taking advantage of Lemma 2.1 and (6) , note that r 1 ≥ 2, we thus have It follows from Lemma 2.1, r 1 ≥ 2 and (6) that To estimate the remaining term R(n, ∇c), we consider the following two cases.
Lemma 2.11. Let 1/r 1 + 1/r 2 = 1, Assume that u ∈ X T and c ∈ Z T . Then we have u · ∇c L 1 where C is a positive constant independent of c and u. In particular, when r 1 = ∞, we have u · ∇c L 1 Proof. The Bony paraproduct decomposition (9) ensures that ∆ j (u∇c) = ∆ j (T u ∇c + R(u, ∇c) + T ∇c u).
[39] Let 1 < p < 6 and f 1 , f 2 being given by (34). We obtain and • The estimate of the pressure P.
Taking div to the first equation of the system (4) yields that By virtue of ∇ · u = 0 and of the notations given by (34), it is clear that and and ∇P L 1 Proof. Applying ∆ j to (43), from L 1 t (L p ) estimates, using (15) from Lemma 2.6, (16) from Lemma 2.7, both (37) and (39) from Lemma 3.1 yield that from which, multiplying by 2 j(−1+3/p) and summing up over j yield the desired result (44). Keeping in minding the proof of (44), applying the operator ∆ j to (42), taking the L 1 t (L p ) norm, using (15) from which, the desired estimate (45) follows readily. Finally, these complete the proof of Proposition 1.
Thanks to [11,41], there exists a positive constantc so that It is clear that, formally, we have the following homogeneous the Bony paraproduct decomposition u∇n λ = T u ∇n λ + R(u, ∇n λ ) + T ∇n λ u.
• The estimate of c.

Now we show Theorem 1.2.
Proof. Under the conditions from Theorem 1.2, proceeding in the same way as [7,35] with minor modifications, there exists a positive time T * , such that the system (4) has a unique local solution (u, n, c) ∈ Θ T * ∩ Θ C T * , where T * is a maximal time of existence, for the more details, we can refer to Appendix A. Hence, to prove Theorem 1.2, we only need to prove that T * = ∞ with (u, n, c) ∈ Θ ∩ Θ C provided that it holds (12).
(100) We can take η and T small enough such that Cη (u 0 , n 0 , c 0 ) E0 ≤ (1−Mi) 2 4K for i = 1 or i = 2, applying Lemma 4.1, we thus get that there exists a positive time T > 0 such that the system (93) has a unique solution (ū,n,c) on [0, T ]. Therefore, applying Proposition 5 yields that the local existence of solution to the system (4). Note that as we have obtain (u, n, c) ∈ Θ T is a solution of (4), then we can proceed in the same way as in the proof of Lemma 2.6-Lemma 2.11, to obtain that u · ∇u, n∇φ ∈ L 1 (0, T ; B −1+3/p p,1 ), u · ∇n, ∇ · (n∇c) ∈ L 1 (0, T ; B −2+3/q q,1 ), u · ∇c, cn ∈ L 1 (0, T ; B 3/q q,1 ). Therefore, it follows from Lemma 2.5 that (u, n, c) ∈ Θ C T . In the following, we will show the solution can be extended from I = [0, T ] to I 1 = [T, T 1 ] for some T 1 > T , here we consider