Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion

In this paper, we study Cauchy problem of the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Taking advantage of a coupling structure of the equations and using the damping effect of the growth term \begin{document}$ g(n) $\end{document} , we obtain the necessary estimates of solution \begin{document}$ (n,c,u) $\end{document} without the diffusion term \begin{document}$ \Delta n $\end{document} . These uniform estimates enable us to establish the global-in-time existence of almost weak solutions for the system.

1. Introduction. The Cauchy problem of the two-dimensional incompressible chemotaxis-Navier-Stokes system with partial diffusion reads , (x, t) ∈ R 2 × R + , ∂ t c + u · ∇c − ∆c = −cn, ∂ t u + u · ∇u − ∆u + ∇P = −n∇Φ, ∇ · u = 0, (n, c, u)| t=0 = (n 0 , c 0 , u 0 ). (1.1) Here n = n(x, t), c = c(x, t) and P = P (x, t) stand for the cell density, certain chemoattractant concentration (for example, oxygen), the pressure of the fluid, respectively. The unknown vector-valued function u(x, t) = u 1 (x, t), u 2 (x, t) represents the velocity field of the surrounding water. In addition, the growth term g(n) is given by g(n) = n(1 − n)(n − a), for 0 < a < 1 2 , which g(n) is also a sufficiently smooth function which related to the cell's growth rate including cooperation and competition effects and the degradation rate due to exterior forces such as predation or intoxication. And the gravitational potential Φ is also a given smooth function. System (1.1) describes a biological process in which cells (e.g. bacteria ) move towards higher concentration of chemically more favorable environment. Some experiments have shown that the mechanism is a chemotactic movement of bacteria often towards higher concentration of oxygen which they consume. At the same time, a gravitational effect on the motion of the fluid is produced by the heavier bacteria, and a convective transport of both cells and oxygen is happened through the water. To know more about its physical background, please see [13,16] for more details.
Due to the significance of the biological background, many mathematicians have studied the similar system (1.2) and made some progress in the past years. Lorz [13] obtained the local existence of the weak solutions for problem (1.2) in a bounded domain in R d , d = 2, 3, with no-flux boundary conditions and the case of inhomogeneous Dirichlet conditions for the oxygen in R 2 . Chae, Kang and Lee [2] proved the local well-posedness and blow-up criterion of smooth solutions of (1.2) in the framework H m with m ≥ 3 in R d , d = 2, 3. Then the result was extended by Zhang to Besov spaces [33].
As for the global-in-time existence of solutions. When κ = 0, a global existence result of weak solutions obtained by Duan, Lorz and Markowich in [3] for problem (1.2) under smallness assumptions on either ∇Φ or the initial data c 0 . The key ingredient of their proof was to establish a priori estimates involving energy type functionals. Subsequently, Liu and Lorz [11] removed this smallness conditions, and obtained the global-in-time existence of weak solutions to the two-dimensional Navier-Stokes version of system (1.2) with κ = 1 for arbitrarily large initial data, under the basically same assumptions on χ and f made in [3]. Recently, Winkler [25] proved that system (1.2) admitted a unique global classical solution in a bounded convex domain Ω with smooth boundary in R 2 under the weaker assumption on χ, f , Φ and initial data than [3,11]. More recently, Zhang and Zheng [35] established some new estimates and proved the global well-posedness of energy solution for the two-dimensional chemotaxis-Navier-Stokes equations in R 2 for the rough initial data.
For the three-dimensional chemotaxis-Navier-Stokes equations, the global classical solutions near constant steady states were constructed in [3] for the system (1.2) with κ = 1. When κ = 0, Winkler [25] showed that problem (1.2), which was considered in a bounded domain and supplemented with the Neumann boundary condition, possessed at least one global weak solution. In [30,31], Winkler further studied existence, eventual regularity and asymptotic stabilization of solutions even for the full chemotaxis-Navier-Stokes equations. However, whether solutions of problem (1.2) with large initial data exist globally or may blow up appears to remain an open problem.
Another case is that ∆n is replaced by the porous medium type expression ∆n m with m > 1 in the first equation of (1.2). We chose m large which should enhance the balance effect of the nonlinear diffusion term, so solutions are likely to remain bounded and global existence. To know more results, we can refer to [3,11,20] for more details.
In 1970, Keller and Segel [8,9] introduced one of the first mathematical models of chemotaxis to describe the aggregation of certain type of bacteria. A simplified version of their model is as follows Many mathematicians have analyzed several different mathematical models of partial differential equations arising in chemotaxis. When g ≡ 0, system (1.3) corresponds to the so-called minimal model Tao and Winkler [19] shown this problem admits at least one global weak solution which approach spatially constant equilibria in the large time limit, under homogeneous Neumann boundary conditions in bounded convex domains. For the 2D homogeneous Neumann problem, Herrero and Velázquez [6] proved that there exist radial solutions that develop a Dirac-delta type singularity in finite time, a feature known in the literature chemotactic collapse. Also they studied the asymptotic of such solutions near the formation of the singularity and the structure of the inner layer around the unfolding singularity. Compared to (1.4), under the assume that f (n, c) = n − c, problem (1.3) comprises a possible proliferation of cells, a growth restriction of logistic type being included by the assumption g(n) ≤ a − µn 2 for all n ≤ 0; accordingly, one might expect that the assumption prevents an unlimited increase of the cell density. This conjecture was supported by numerical experiments, which indicated that (1.3) possessed quite a large variety of dynamical properties, especially in respect of the spontaneous emergence of patterns, though apparently simple as a two-component parabolic system. Further evidence, both numerically and analytically, on the self-organizing features of (1.3) can be found in [16], where shock-type movements of interfaces are detected as g(n) = n(1 − n)(n − a), 0 < a < 1 2 for cell kinetics take place much faster than cell movement [5]. For the other interesting results, one can refer [10,22,24,26,27] for more detail.
The present paper is mainly devoted to study the global existence of weak solution to system (1.1) for initial data. Now we state our main result. For clarity, we set Here the weak solution defined as follows: Definition 1.2 (Weak solution). We call (n, c, u) is a global-in-time weak solution of system (1.1) if for any T > 0, The triple (n, c, u) satisfies equations (1.1) in the sense of distribution.
Remark 1. Compared with (1.2), we obtain the global existence of weak solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Until now, there are few mathematicians study the model without diffusion term ∆n. In our paper, the growth term g(n) ensures us to obtain the necessary estimates we need, even without the term ∆n. Taking advantage of a coupling structure of the equations, using the technology of double regularization and energy estimate, we explore the related estimates of approximate solutions. After that by establishing a priori estimates and using the Arzelá-Ascoli theorem, we prove the global existence of weak solutions for the system we studied. Remark 2. It seems, however, that our methods can readily be adapted to similar model by slightly modifying the proof of uniform estimates in section 4. In fact, one could conclude the so-called "energy-type inequality" which is the key ingredient of our proof, by combining our approach and the double regulariization technology in [14].
The paper is organized as follows. In Section 2, we review the theory of Littlewood and Paley operator, the norm of two kinds of mixed space-time Besov space, and the related lemma which will be used in the following sections. In Section 3, we study the global well-posedness for the regularized problem. We prove the existence of global classical solution (n ε , c ε , u ε ) for such a problem, which relying on some results presented in the recent literature. Section 4 is devoted to showing uniform estimates for the regularized problem which is the heart of our analysis, and we will provide ε-independent estimates for this classical solutions, which are attained by means of functional and algebra inequalities. The proof of the main result will be given in the last section.
Notation. Throughout the paper, R + = (0, ∞) and C stands for a "generic" positive constant which may changes from line to line. For p, q ∈ [0, ∞], the usual Lebesgue space is denoted by L p (R 2 ) and · L q t L p denotes the norm of is made up of all continuous function f (x) on R 2 with |α| ≤ k order continuous partial derivative ∂ α f (x), and space S(R 2 ) is the Schwartz class of rapidly decreasing functions.

2.
Preliminaries. In this preparatory section, we provide the definition of some function spaces and review some important lemmas. Here, we recall the theory of Littlewood-Paley which is useful to measure smoothing efforts of the linear heat equation. To define Besov Spaces, we start with the dyadic decomposition.
We define also the function χ(ξ) = 1 − q∈N ψ(2 −q ξ). For u ∈ S R 2 ), the inhomogeneous dyadic blocks are defined by: We also introduce the following low-frequency cutoff In terms of Littlewood-Paley operator ∆ j , we can define inhomogeneous Besov spaces as follows. Let (p, r) ∈ [1, +∞] 2 and s ∈ R, then the inhomogeneous space B s p,r (R 2 ) is the set of tempered distributions u such that: It is worthwhile to remark that B s 2,2 and B s ∞,∞ coincide with the usual Sobolev spaces H s and the usual Hölder space C s for s ∈ R + \Z respectively. For more details, we can refer to [1,15].
In our next study, we require two kinds of mixed space-time Besov space. The first one is defined in the following manner: for T > 0 and ρ ≥ 1, we denote by L ρ T B s p,r the set of all tempered distributions u satisfying The second mixed space is L ρ T B s p,r , which consists of tempered distributions u satisfying And the norm · L ρ T is defined as following Minkowski's inequality entails that if s ∈ R, ρ ≥ 1, and (p, Finally, we would recall some useful lemmas which are powerful tools in the proof of our main result: Lemma 2.2 ( [15]). There exists a constant C depending only on ϕ and such that for all q ∈ Z, λ ≥ 0, p ∈ [1, +∞], and u ∈ S , we have

Lemma 2.3 ([14]
, Sobolev inequality). The space H s+k (R 2 ), s > 1, k ∈ Z + ∪ {0}, is continuously embedded in the space C k (R 2 ). That is, there exists C > 0 such that (2.1) 3. Solutions to the regularized problem. This section is to modify equations in order to produce a family of global smooth solutions. We begin by a regularizing operator called a mollifier. Given any radial function define the standard mollifier Now let us consider the regularized system governed by where g(n ε ) = (1 + a)(n ε ) 2 − an ε − (n ε ) 3 . We first establish the global existence of smooth solutions of (3.1), by using energy estimates and classical compactness argument.
Assume, in addition, that the initial data n ε 0 and c ε 0 are positive. Then the regularized system (3.1) admits a unique global solution And we define the convolution operator J k as follows: . We are going to construct the following approximate system: Following Leray operator, we eliminate the pressure P k,ε and the incompressibility condition div u k,ε = 0 by protecting the third equation of the above system onto the space of divergence-free functions Then, problem (3.2) reduces to an ODE in the Banach Space H s,σ (R 2 ): Let us imitate its relation proof process, we could know that problem (3.2) and (3.3) are equivalent. We can refer to [15] for the detailed proof.
Step 1. We first show the existence and uniqueness of solution of (3.2) for each k ∈ N + . Specially, we have the following proposition: Proof of Proposition 3.2. Firstly, we are going to apply the Picard theorem to get a local-in-time solution of problem (3.3). It suffices to show that for each k ≥ 1 and Because the proofs of (3.4), (3.5) and (3.6) are analogous, we just need to show the first inequality (3.4). Based on the Morse estimate (2.1), simple calculations yield where By Lemma 2.1, the estimate (2.1) and the properties of mollifier, we see that

Similarly, we get that
(3.8) On one hand, On the other hand, we see that Those estimates together with (3.8) yield A similar estimate holds for I 3 -I 6 , and finally, we show that Collecting these estimates leads to Thanks to estimates (3.4) (3.5) and (3.6), we get that F k,ε maps H s (R 2 ) into H s (R 2 ) and F k,ε is locally Lipschitz continuous on any open set Hence, by Picard Theorem [14, Theorem 3.1], we have that for every ( We finish the proof of the local existence.
Step 2. In this step, we are devoted to proving that there exists T > 0 such that T k ≥ T for all k ∈ N + . Thanks to blow up criterion, it is suffice to show the following proposition.
Proof of Proposition 3.3. Taking the L 2 -inner product with the second equation of (3.2) with c k,ε yields that The properties of J k and ρ ε together with the fact (n k,ε , c k,ε , u k,ε ) allows us to perform integrations by parts without a boundary term in what follows. With this in hand, integrations by parts lead to Inserting the above equality into (3.11) yields (3.12) Taking the L 2 -inner product with the first equation of (3.2) with n k,ε , we have (3.13) Obviously, by the Hölder inequality and the Young inequality, we have (3.14) For K 2 , by the Interpolation inequality that and the ε-Young inequality, we obtain that Plugging (3.14) and (3.15) into (3.13), we have By the standard energy estimate, we can conclude that for s > 1 We finish the proof of the local-in-time unifom estimate independent of k.
Step 3. Our target now is to prove that system (3.1) admits a local-in-time solution.
This obviously entails that (n k,ε , c k,ε , u k,ε ) tends to (n ε , c ε , u ε ) in D (R + ×R 2 ). The Fatou's lemma ensures that (n ε , c ε , We still have to prove that (n ε , c ε , u ε ) is continuous in H s × H s × H s . Let us apply the operator ∆ q (q ≥ 0) to the first equation of (3.1), we get Taking the L 2 -inner product with the above equation with ∆ q n ε , yields So, the Hölder inequality and the Young inequality ensure us to have It implies that Multiplying the above inequality by 2 2qs and taking the l 2 -norm imply This is to say, the sequence {S N n ε } N ∈Z + converges uniformly to n ε in L ∞ t H s . On the other hand, the fact ∂ t n ε ∈ L 2 loc (R + ; H s−1 (R 2 )) allows us to conclude that n ε ∈ C(R + ; H s (R 2 )) with s < s. This means that S N n ε ∈ C(R + ; H s (R 2 )) for a fixed N ∈ Z + . As a result, we obtain n ε ∈ C(R + ; H s (R 2 )). By the same argument with n ε , we can conclude that c ε ∈ C(R + ; H s (R 2 )) and u ε ∈ C(R + ; H s (R 2 )). Now we need to show the positivity of n ε > 0, c ε > 0 for all (x, t) ∈ R 2 × [0, T ]. Let us set (n ε ) − min{n ε , 0}.
Multiplying the first equation of (3.1) by (n ε ) − and then integrating in space variable x, we readily have Integrating by parts and using the Hölder inequality, one has On the other hand, we see that The Grönwall inequality implies that for any t ≤ T (n ε ) − (t) L 1 ≤ (n ε 0 ) − L 1 C(t) = 0, which means n ε ≥ 0 for almost everywhere (x, t) ∈ R 2 × [0, T ]. Since s > 1, we have H 1 (R 2 ) → C b (R 2 ). Hence we have that n k,ε ≥ 0 for all (x, t) ∈ R 2 × [0, T ]. By the same argument, we can get the positivity of c k,ε ≥ 0.
Step 4. In this step, we shall establish the global-in-time H s -estimates for (n ε , c ε , u ε ). Taking the L 2 -inner product with the second equation of (3.1) with c ε yields that 1 2 Integrations by parts lead to Inserting the above equality into (3.16) and using n ε ≥ 0, one has which implies that Taking the L 2 -inner product with the first equation of (3.1) with n ε , we have (3.18) Obviously, by the Hölder inequality and the Young inequality, we have For K 2 , by the interpolation inequality that and the ε-Young inequality, we obtain that we have used inequality (3.17) in the last line. Taking the L 2 -inner product with the third equation of (3.1) with u ε implies (3.23) Next, we will show the H 1 -estimate of solution (n ε , c ε , u ε ). Based on the equation multiplying the above equation by −∆n ε and integrating with respect to space variable x yields that

(3.24)
Clearly, the term L 1 can be bounded by using the Hölder inequality and the Young inequality, as follows For the term L 2 , the Hölder inequality and the Young inequality allow us to infer that By the integration by parts and simple calculations, we know Plugging the estimates of L 1 -L 3 into (3.24) leads to (3.25) Performing the same argument as above, we get that Taking the curl to the third equation of (3.2), multiplying the resulting equation by ω ε and integrating with respect to the space variable mean that (3.27) Summing up the estimates (3.25), (3.26) and (3.27), then by the Grönwall inequality, we obtain the H 1 norm of n ε , c ε , ω ε are all closed. The expression is as follows Combining this with (3.23) yields Last, we shall give the H s -norm for solutions by using the Fourier localization technique. Apply ∆ q to the first equation of (3.2) and then multiplying the resulting equality by ∆ q n ε , we have where Now, we shall estimate each term on the right hand side of (3.28). Firstly, for the term M 1 and M 2 , by integration by parts, the Hölder inequality and the Young inequality, we directly get that and (3.30) Similarly, for the term M 3 and M 4 , we obtain and Plugging (3.29)-(3.32) into (3.28), then we have the following estimate . Now, multiplying both sides of the above inequality by 2 2qs , and then computing the 1 -norm, we have 1 2 By Leibniz estimate (2.1), we know that So, (3.33) Similarly, applying ∆ q to the second equation of (3.2) and then multiplying the resulting equality by ∆ q c ε yields that By the Hölder inequality and the Young inequality, we have Multiplying 2 2qs on both sides of the above inequality, then taking the l 1 -norm, we obtain d dt (3.34) Apply ∆ q to the third equation of (3.2) and then multiplying the resulting equality by ∆ q u ε implies Similarly, by the Hölder inequality and the Young inequality, we get The above equation implies that Multiplying 2 2qs on both sides of the above inequality, then taking the 1 -norm, we get Then we obtain Step 5. This step is devoted to proving the uniqueness of solutions. Let (n ε i , c ε i , u ε i ) i = 1, 2 be two solutions of the system (3.1) with the same initial dada (n ε 0 , c ε 0 , u ε 0 ). Denote δn ε = n ε 1 − n ε 2 , δc ε = c ε 1 − c ε 2 , δu ε = u ε 1 − u ε 2 . Then we have the difference 3432 LAIQING MENG, JIA YUAN AND XIAOXIN ZHENG equations as follows: where H(x, t) −∇ · δn ε ∇(c ε 1 * ρ ε ) − ∇ · n ε 2 ∇(δc ε * ρ ε ) + g(n ε 1 ) − g(n ε 2 ), and Taking the L 2 -inner product with the first equation with δn ε , we have (3.38) where ∇ · (n ε 2 ∇(δc ε * ρ ε ))δn ε dx, Now we need to estimate the above equations, respectively. For the first term O 1 , we have (3.39) Similarly, we have Applying the integration by parts, the Hölder inequality and the Young inequality, we get and as well as (3.43) Let's just assume that n ε 1 < n ε 2 , then Similarly, we can infer that Summing up the above three equations, we have Consequently, we get Therefore, we obtain the uniqueness by using the Grönwall inequality on time [0, T ]. Lastly, we wan to show that (n ε , c ε , we can show by the Leibniz estimates that for s > 1 It follows from argument in [14,Theorem 3.5] that ∂ t u ε ∈ C(0, T ; H s−2 ). The proof of Proposition 3.1 is completed.

4.
Uniform estimate for the regularized problem. In this section, we prove the uniform estimates for smooth solutions (n ε , c ε , u ε ) to the regularized problem (3.1) which are independent neither of ε > 0 nor the mollifier ρ ε . By Proposition 3.1, we know that the regularized system (3.1) admits a unique global solution (n ε , c ε , u ε ) ∈ C [0, ∞); H s (R 2 ) ∩ L 2 loc [0, ∞); H s+1 (R 2 ) 3 . On the other hand, for l > 1 space H l (R 2 ) is a Banach algebra embedded in the set of continuous functions going to 0 at infinity. It means that the solutions to the regularized problem (3.1) are smooth and decay sufficiently fast at infinity, so when we integrate by parts in our calculations below, there are no boundary terms. We distinguish especially two kinds: the first one deals with some easy estimates that one can obtained by energy estimates. The second one is concerned with some strong estimates which are the heart of the proof of our main result.
Proposition 4.1. Let the triple (n ε 0 , c ε 0 , u ε 0 ) ∈ X 0 ∩ (H s (R 2 )) 3 with s > 1 and ∇Φ ∈ L ∞ . Assume (n ε , c ε , u ε ) is a smooth solution of system (3.1). Then there exists a constant C > 0 independent of ε > 0 such that Proof. Since n ε 0 ≥ 0 and c ε 0 ≥ 0, we know from Proposition 3.1 that n ε (x, t) > 0 and c ε (x, t) > 0 for all (x, t) ∈ R 2 × R + . Firstly, we deal with the first equation of (3.1). Since ∇ · u ε = 0, we have d dt n ε (t) L 1 + a n ε (t) L 1 + n ε (t) 3 then we have d dt n ε (t) L 1 + a n ε (t) Hence, by the Grönwall inequality, we get that Then we tackle the second inequality. Multiplying the second equation of (3.1) by (c ε ) p−1 (2 ≤ p < ∞) and integrating the resulting equation, yields Then we get So, by the Grönwall inequality, we obtain Lastly, multiplying both sides of the third equation of (3.1) by u ε , we have (4.2) The above inequality implies that By the Grönwall inequality and the Interpolation theorem, we directly know that where we have used estimate (4.1). Plugging (4.3) into (4.2), we have By the Grönwall inequality, the Hölder inequality and the Interpolation theorem, we obtain that This completes the proof of Proposition 4.1. 3 with s > 1 and ∇Φ ∈ L ∞ . Assume (n ε , c ε , u ε ) is a smooth solution of system (3.1). Then there exists a constant C > 0 depending only on the initial data such that Proof. Multiplying n ε on both sides of the first equation of (3.1), and integrating the resulting equation yields By the Hölder inequality, the Interpolation theorem and the Young inequality, we find that the above equation can be bounded as following Now, we turn to the second equation of (3.1), multiplying it by −∆c ε and integrating the resulting equation, we could get Similarly, by the Hölder inequality and the Young inequality, we obtain from which we have 1 2 d dt ∇c ε (t) 2 L 2 + 3 4 ∆c ε (t) 2 L 2 ≤ C ∇u ε (t) 2 L 2 ∇c ε (t) 2 L 2 + C c ε (t) 2 ∞ n ε (t) 2 L 2 . (4.5) Next, we shall consider the last one. According to the third equation of (3.1), ω = ∂ 1 u 2 − ∂ 2 u 1 solves ∂ t ω ε + (u ε · ∇)ω ε − ∆ω ε = −∂ 1 (n ε ∂ 2 Φ) + ∂ 2 (n ε ∂ 1 Φ).
Similarly, we claim that ∂ t c ε is bounded in L 2 loc (R + ; By the same argument with ∂ t c ε , we can infer that ∂ t u ε is bounded in L 2 loc (R + ; H −1 ). We know that L 2 local compactly embeds in H s and H s continuously embed in H −1 with s ∈ (−1, 0). Thanks to Proposition 4.1 and Proposition 4.2, we we know that