TIME PERIODIC SOLUTION TO A COUPLED CHEMOTAXIS-FLUID MODEL WITH POROUS MEDIUM DIFFUSION

. This paper is concerned with the time periodic problem to a cou- pled chemotaxis-ﬂuid model with porous medium diﬀusion ∆ n m . The global existence of solutios for the initial and boundary value problem of this model have been studied by many authors, and in particular, the global solvability is established for m > 65 in dimension 3. Here, taking advantage of a double-level approximation scheme, we establish the existence of uniformly bounded time periodic solution for any m ≥ 65 and any large periodic source g ( x,t ). In par- ticular, the energy estimates techniques we used also applicable to the proof of global existence of the initial-boundary value problem, and one can supply the existence of global solutions for m = 65 by this method.

The following coupled chemotaxis-fluid model with consumption was first proposed by Tuval, Goldstein et. al. in 2005, n t + u · ∇n = ∆n − ∇ · (n∇c), c t + u · ∇c = ∆c − cn, u t + ku · ∇u = ∆u − ∇π + n∇ϕ, , ∇ · u = 0, which describes the dynamics of bacterial swimming and oxygen transport near contact lines. Since then, the coupled chemotaxis-fluid models have been studied by many authors. In two dimensional bounded domain, Winkler [20,22] established the global existence and stability of classical solutions. Whereas, in three dimensional case, the result on global existence of classical solutions is not perfect, there are only some global existence results on small initial data [2,15], and for the study of weak solutions, please refer to [12,20,23,24]. Beside these results, there are also many papers that concerned with the chemotaxis-fluid model with production. That is the model (1.1) with m = 1. Espejo, Tao, et.al [6,18] proved the global existence of weak solutions and classical solutions respectively in two dimensional settings. For the three dimensional case, in [17], Tao and Winkler established the global solvability of classical solutions for large µ, and in [21], a global weak solution is achieved for any positive µ. While as far as the porous medium diffusion case m > 1, only a few works involves this aspect of research. Liu, Wang [13] considered the following system in dimension 3          n t + u · ∇n = ∆n m − χ∇ · (n(1 + n) −α ∇c) + µn(1 − n), c t + u · ∇c = ∆c − c + n, u t = ∆u − ∇π + n∇ϕ, ∇ · u = 0, and proved the existence of global bounded weak solution for α > 0, m ≥ 1 3 , α+m > 6 5 . While, if consider the fluid free case , that is let u ≡ 0. Then the problem (1.1) is transformed into the pure chemotaxis system. n t = ∆n m − χ∇ · (n∇c) + µn(1 − n), c t = ∆c − c + n.
When µ = 0, it has been shown that when N ≥ 2, there exists a threshold value m * = 2N −2 N such that the solutions always exists globally when m > m * , and blow up solutions will be generated for some initial values when m ≤ m * . Please refer to [3,4,5,10] and the references therein for more detais. While, when µ > 0, although there is no direct research result on this model, there are some related works on the coupled chemotaxis-haptotaxis model. However, if we ignore the haptotaxis term, it happens to be the pure chemotaxis model above. Through the efforts of many researchers [16,11,19,25], it finally proved the global solvability of bounded weak solutions for m > 2N N +2 . However, the global existence of bounded weak solutions for the case m ≤ 2N N +2 remains unknown. While as far as the time periodic solutions are concerned, there are few works concerned. In 2017, Jin [7,8] considered the time periodic problem for the linear diffusion case of (1.1) in dimension 2 and dimension 3, namely the case m = 1, and proved the existence of time periodic solutions. In the present paper, we pay our attention to the time periodic problem (1.1) in three dimensional case. We use a double-level approximation scheme based on a fourth order regularized system. To obtain the compactness of the operator, we use the first-level fourth order regularized system to approach the original system. However, different from the second order parabolic system, there is no positivity for the fourth order regularized system. So, the L 1 -norm estimate can not be obtained directly, which brings great difficulty to the later proof. To solve this problem, we add a term ε|n| s n to the left side of the first equation with s appropriately large. And in the second level approximation, we use some iterative techniques to obtain some uniform energy estimates and finally showed the existence of time periodic solutions for any m ≥ 6 5 . In the second level approximation, the energy estimates techniques we used also applicable to the proof of global existence of the initial-boundary value problem, in particular, using this method, one can supply the existence of global solutions for m = 2N N +2 . More precisely, we have the following result.

2.
Preliminaries. For reader's convenience, we give some notations, which will be used throughout this paper.
To prove the existence of time periodic solutions, we begin with some preliminary lemmas.
By [9], we have the following two lemmas Lemma 2.1. Let T > 0, a > 0, ≥ 0, and suppose that f : R + → [0, ∞) is absolutely continuous, f, h are time periodic functions with period T , and f satisfies Lemma 2.2. Let T > 0, a > 0, > 0, and suppose that f : R + → [0, ∞) is absolutely continuous, f, g, h are time periodic functions with period T , and satisfies Then sup t∈(0,T ) where C is a constant depending only on a, α, β, T . While, if a = 0, and We also have that where C is a constant depending only on γ, α, β, T .
admits a unique strong time periodic solution u ∈ W 2,1 where C is a positive constant.
The following lemma follows from [14].
3. Time periodic solutions for a fourth-order regularized problem. In this paper, we use a double-level approximation scheme to show the existence of time periodic solutions. To obtain the compactness of the operator, in the first level, we use a fourth order regularized system as follows to approach the original system.
where max{2(m − 1), 4} < s ≤ 5m − 1. It is worth noticing that althrough the four order term can improve the regularity of solutions, there is no positivity for the fourth order regularized problem. Therefore, the most basic and natural L 1 -norm estimate of n is no longer valid, which brings essential difficulties to the later proof of energy estimates, see for example the proof of (3.9), (3.13) etc. For this reason, we introduce the term ε|n| s n to solve the difficulties caused by the lack of positivity. To prove the existence of time periodic solutions for the problem (3.1), we linearize this problem. For u, consider the following linear problem, where ∈ [0, 1] is a constant. Taking advantage of [7,8], we have Lemma 3.1. Assume thatn ∈ L 2 T (R, L 2 (Ω)). Then the problem (3.2) admits a time periodic solution u ∈ L ∞ ((0, T ), H 1 σ (Ω) ∩ L 2 ((0, T ), H 2 σ (Ω)), and u t ∈ L 2 ((0, T ), L 2 σ (Ω)).
For the above obtained solution u, let's consider the following linear problem. 3) The following lemmas follows from [7,8]. For the above obtained solutions u, c, consider For the above linear parabolic problem, when A is sufficiently large, the existence of time periodic solutions can be easily obtained by a fixed point method. That is, define a Poincaré map from n(x, 0) to n(x, T ), the time-periodic solution is then identified as a fixed point of this Poincaré map. We only give the regularity estimates. For simplicity, in what follows, we may assume that the solution n is sufficiently smooth, otherwise, we can approximate u, c,n, g with a sequence of sufficiently smooth functions u k , c k ,n k , g k such that the corresponding solutions n are sufficiently smooth, and the following energy estimates can be obtained through an approximate process. Proof. Multiplying the first equation of (3.4) by n,integrating it over Ω × (t 0 , t) for any t 0 < t ≤ t 0 + T , and using Lemma 3.2, when A sufficiently large, we see that We multiply the first equation of (3.4) by ∆n, and integrate it over Ω × (t 0 , t) with any t 0 < t ≤ t 0 + T to obtain Combining with (3.5), and using Lemma 2.1, we get that Multiplying the first equation of (3.4) by ∆ 2 n, integrating it over Ω × (t 0 , t) for any t 0 < t ≤ t 0 + T , and using Lemma 3.1, Lemma 3.2 and (3.6), we get combining with (3.6), we finally get Similar to the proof of the above formula, multiplying the first equation of (3.4) by n t and integrating it over Ω, it follows Summing up, we complete the proof. Below we consider the existence of time periodic solutions to the problem (3.1). Denote where n is the solution of the problem (3.8). Next, we use Leray-Schauder's fixed point theorem to prove the existence of time periodic solutions to the problem (3.1).
For this purpose, we first give the below priori estimate.
, and let F(n, ) = n. Then there exists R > 0 such that where R depends on ε, δ, and is independent of A.
Proof. Takingn = n in (3.4). Multiplying the first equation of (3.4) by n, and integrating it over Ω × (t 0 , t) for any t 0 < t ≤ t 0 + T , we see that which implies that Takingn = n in (3.2) and multiplying the corresponding equation by 2u, 2∆u respectively, then combining the two inequalities, it is easy to see that 3), and multiplying this equation by c, ∆c respectively, and integrating them over Ω × (t 0 , t) for any t ≥ t 0 , we have and Combining (3.9) and (3.10), and taking advantage of Lemma 2.1, we obtain that (3.12) Recalling (3.9), using (3.12), and noticing that s > 4, we arrive at sup t∈(0,T ) Ω where C ε is independent of δ and A, but depends on ε. By (3.13), recalling Lemma 3.2, we also have sup t∈(0,T ) Ω (3.14) Here C ε is also independent of δ and A, but depends on ε. Takingn = n in (3.4).
Multiplying the first equation of (3.4) by −∆n , integrating it over Ω × (t 0 , t) for any t 0 < t ≤ t 0 + T , and using (3.10) and (3.13), noticing that s > 2(m − 1), then we see that t0 Ω |∇∆n| 2 dxds + t t0 ∆n L 2 n∆c + ∇n∇c L 2 ds + C By (3.14), and we see that where C εδ depends on ε and δ, and independent of A. By (3.13) and (3.15), we complete the proof. By Lemma 3.3, we see that F is a compact operator. Furthermore, by (3.9), it is easy to see that F(n, 0) = 0.
By Lemma 3.4, we see that if F(n, ) = n, then there exists a constant M ≥ 0, such that n X ≤ M, where M depends on ε, δ. Then by Leray-Schauder's fixed point theorem, the mapping F(·, 1) has a fixed point, that is there exists n ∈ X such that F(n, 1) = n, that is the problem (3.1) admits a solution (n, c, u). Furthermore, taking advantage of Lemma 3.3, we have the following proposition.

4.
Existence of time periodic solutions: Double-level approximation. In this section, we use a double-level approximation scheme to prove the existence of time periodic solutions of the problem (1.1). We consider the first level approximation by letting δ → 0. For this purpose, we first give some energy inequalities. Taking advantage of (3.10)-(3.14), we have Lemma 4.1. Assume that s > max{2(m−1), 4}, g, ∇ϕ ∈ L ∞ T (Q), and let (n εδ , c εδ , u εδ , π εδ ) be the periodic solution of the problem (3.1). Then we have that Here C is a constant not depending on δ, only depending on Ω, T, g, ∇ϕ, and ε.
To show this proposition, we first have the following lemma. For simplicity, we still denote the solution of (4.4) by (n, c, u, π). Lemma 4.3. Assume g ≥ 0 with g, ∇ϕ ∈ L ∞ T (Q). Let (n, c, u, π) be a time periodic solution of (4.4). Then we have n ≥ 0, c ≥ 0 and where these constants C are independent of ε.
To improve the regularity of the solutions, we first introduce the below Lemma.
Lemma 4.5. Assume m > 1, g ≥ 0 with g, ∇ϕ ∈ L ∞ T (Q). Let (n, c, u, π) be a time periodic solution of (4.4). Assume that there exists q > 2, such that Here these constants C are independent of ε.
Proof of Proposition 4.1. Multiplying the first equation of (4.4) by 1 + ln n, and integrating it over Ω × (t 0 , t) for any t 0 < t ≤ t 0 + T , we see that