Existence of global solutions to chemotaxis fluid system with logistic source

We establish the existence of global solutions and Lq time-decay of a three dimensional chemotaxis system with chemoattractant and repellent. We show the existence of global solutions by the energy method. We also study Lq time-decay for the linear homogeneous system by using Fourier transform and finding Green’s matrix. Then, we find Lq time-decay for the nonlinear system using solution representation by Duhamel’s principle and time-weighted estimate.


Introduction
Chemotaxis is the oriented movement of biological cells or microscopic organisms toward or away from the concentration gradient of certain chemicals in their environment. We may use cells to denote the biological objects whose movement we are interested in and chemo attractants or repellents to denote chemicals which attract or repell the cells. This type of movement exists in many biological phenomena, such as the movement of bacteria toward certain chemicals [1], or the movement of endothelial cells toward the higher concentration of chemoattractant that cancer cells produce [4]. Keller and Segel [11,12] derived a mathematical model to describe the aggregation of certain types of bacteria, which consists of the equations for the cell density n = n(x, t) and the concentration of chemical attractant c = c(x, t) and is given by n t = ∆n − ∇ · (nχ∇c), where χ is the sensitivity of the cell movement to the density gradient of the attractant, α is a positive constant, and the reaction term f is a smooth function of the arguments. Since then, many mathematical approaches to describe chemotaxis using systems of partial differential equations have emerged, some of which will be discussed later in this section.
In this paper, we use the equations for continuum mechanics to describe the movement of cells and for the chemoattractant and repellent, we use diffusion equations. The combined effects of chemoattractant and repellent for chemotaxis are studied in diseases such as Alzheimer's disease [2].
We consider the initial value problem for the system in R 3 given by ∂ t u + u · ∇u + ∇p(n) n = χ 1 ∇c 1 − χ 2 ∇c 2 + δ∆u ∂ t c 1 = ∆c 1 − a 12 c 1 + a 11 c 1 n ∂ t c 2 = ∆c 2 − a 22 c 2 + a 21 c 2 n, where n(x, t), u(x, t), c 1 (x, t), c 2 (x, t) for t > 0, x ∈ R 3 , are the cell concentration, velocity of cells, chemoattractant concentration, and chemorepellent concentration, respectively. The initial data is given by (n, u, c 1 , c 2 )| t=0 = (n 0 , u 0 , c 1,0 , c 2,0 )(x), where it is supposed to hold that (n 0 , u 0 , c 1,0 , c 2,0 )(x) → (n ∞ , 0, 0, 0) as |x| → ∞, for some constant n ∞ > 0. In this model the cells follow a convective logistic equation, the velocity is given by the compressible Navier-Stokes type equations with the added effects of chemoattractants and -repellents. The pressure for the cells p(n) is a smooth function of n and p (n) > 0, a positive constant δ is the coefficient for the viscosity term, and χ 1 and χ 2 express the sensitivity of the cell movement to the density gradients of the attractants and repellents, respectively. Usually χ i , (i = 1, 2) are functions of c i and in this paper we consider the case χ i = K i c i , where K i are positive constants, so that the sensitivity is proportional to the concentration of the attractants and repellents. We choose K i = 2 for simplicity. We may equally use χ i = K i c α i i , where α i are positive constants. For chemical substances, we use the reaction diffusion equations. The reaction terms are based on a Lotka-Volterra type model in which the nonnegative regions of c i are invariant in the sense that if the initial conditions for c i are nonnegative, they are nonnegative for positive t. This can be verified by the maximum principle. The couplings between c i and n are given as nonlinear terms.
The main goal of this paper is to establish the local and global existence of smooth solutions in three dimensions around a constant state (n ∞ , 0, 0, 0) and the decay rate of global smooth solutions for the above system (1.1). The main result of this paper is stated as follows. and there are constants λ 1 > 0 and λ 2 > 0 such that Furthermore, the global solution [n, u, c 1 , c 2 ] satisfies the following time-decay rates for t ≥ 0: The proof of the existence of global solutions in Theorem 1.1 is based on the local existence and an a priori estimates. We show the local solutions by constructing a sequence of approximation functions based on iteration. To obtain the a priori estimates we use the energy method. Moreover, to obtain the time-decay rate in L q norm of solutions in Theorem 1.1, we first find the Green's matrix for the linear system using the Fourier transform and then obtain the refined energy estimates with the help of Duhamel's principle.
To motivate our study, we present previous related work on chemotaxis models. Many of them are based on the Keller-Segel system. Wang [21] explored the interactions between the nonlinear diffusion and logistic source on the solutions of the attraction-repulsion chemotaxis system in three dimensions. E. Lankeit and J. Lankeit [13] proved the global existence of classical solutions to a chemotaxis system with singular sensitivity. Liu and Wang [14] established the existence of global classical solutions and steady states to an attraction-repulsion chemotaxis model in one dimension based on the energy methods.
Concerning the chemotaxis models based on fluid dynamics, there are two approaches, incompressible and compressible. For the incompressible case, Chae, Kang and Lee [3], and Duan, Lorz, and Markowich [8] showed the global-in-time existence for the incompressible chemotaxis equations near the constant states, if the initial data is sufficiently small. Rodriguez, Ferreira, and Villamizar-Roa [19] showed the global existence for an attractionrepulsion chemotaxis fluid model with logistic source. Tan and Zhou [20] proved the global existence and time decay estimate of solutions to the Keller-Segel system in R 3 with the small initial data. For the compressible case, Ambrosi, Bussolino, and Preziosi [2] discussed the vasculogenesis using the compressible fluid dynamics for the cells and the diffusion equation for the attractant.
Many related approaches use the Fourier transform, and we only mention that Duan [6] and Duan, Liu, and Zhu [7] proved the time-decay rate by the combination of energy estimates and spectral analysis. Also by using Green's function and Schauder fixed point theorem, one can study the existence and regularity of solution for these kinds of equations (see [9,10,17,18]).
For later use in this paper, we give some notations. C denotes some positive constant and The length of α is |.| = α 1 + α 2 + α 3 ; we also set ∂ j = ∂ x j for j = 1, 2, 3. For an integrable x j ξ j , and x ∈ R 3 , where i = √ −1 is the imaginary unit. Let us denote the space This paper is organized as follows. In Section 2, we reformulate the Cauchy problem under consideration. In Section 3, we prove the global existence and uniqueness of solutions. In Section 4, we investigate the linearized homogeneous system to obtain the L 2 − L q timedecay property and the explicit representation of solutions. In Section 5, we study the L q time-decay rates of solutions to the reformulated nonlinear system and finish the proof of Theorem1.1.
In what follows, the integer N ≥ 4 is always assumed.
The proof of Theorem 1.1 obtained directly from the global existence proof in Proposition 2.1 and the derivation of rates in Theorem 1.1 is based on Proposition 2.2.

Global solution of the nonlinear system (2.2)
The goal of this section is to prove the global existence of solutions to the Cauchy problem (2.2) when initial data is a small, smooth perturbation near the steady state (n ∞ , 0, 0, 0). The proof is based on some uniform a priori estimates combined with the local existence, which will be shown in Subsections 3.1 and 3.2.

Existence of local solutions
In this subsection, we show the proof of the existence of local solutions [ρ, u, c 1 , c 2 ] by constructing a sequence of functions that converges to a function satisfying the Cauchy problem. We construct a solution sequence (ρ j , u j , c j 1 , c j 2 ) j≥0 by iteratively solving the Cauchy problem on the following as |x| → ∞, for j ≥ 0. For simplicity, in what follows, we write U j = (ρ j , u j , c j 1 , c j 2 ) and U 0 = (ρ 0 , u 0 , c 1,0 , c 2,0 ), where U 0 = (0, 0, 0, 0). Now, we can start the following Lemma. Lemma 3.1. There are constants T 1 and 0 > 0 such that if the initial data U 0 ∈ H N (R 3 ) and U 0 H N ≤ 0 , then there exists a unique solution U = (ρ, u, c 1 , c 2 ) of the Cauchy problem (2.2)-(2.3) on [0, T 1 ] with U ∈ X(0, T 1 ).
Proof. We first set U 0 = (0, 0, 0, 0). Then, we use U 0 to solve the equations for U 1 . The first equation is the first order partial differential equation and the second, third, and fourth equations are the second order parabolic equations. We obtain u 1 (x, t), c 1 1 (x, t), c 1 2 (x, t), and ρ 1 (x, t) in this order. Similarly, we define (u j , c j 1 , c j 2 , ρ j ) iteratively. Now, we prove the existence and uniqueness of solutions in space C([0, T 1 ]; H N (R 3 )), where T 1 > 0 is suitably small. The proof is divided into four steps as follows.
In the first step, we show the uniform boundedness of the sequence of functions under our construction via energy estimates. We show that there exists a constant M > 0 such that for all j ≥ 0. We use the induction to prove (3.3). It is trivial when j = 0. Suppose that it is true for j ≥ 0 where M is small enough. To prove for j + 1, we need some energy estimate for U j+1 . Applying ∂ α to the first equation of (3.1), multiplying it by ∂ α ρ j+1 and integrating in x, we obtain The terms on the right hand side are further bounded by Then, after taking the summation over |α| ≤ N and using the Cauchy inequality, one has Similarly, applying ∂ α to the second equation of (3.1), multiplying it by ∂ α u j+1 , taking integrations in x, and then using integration by parts, we have Then, after taking the summation over |α| ≤ N, the terms on the right side of the previous equation are bounded by By using the Cauchy inequality, we obtain In a similar way as above, we can estimate c 1 and c 2 as Taking the linear combination of inequalities (3.4)-(3.7), we have Thus, after integrating with respect to t, we have In the last inequality, we use the induction hypothesis. We obtain for 0 ≤ t ≤ T 1 . This implies that (3.3) holds true for j + 1. Hence (3.3) is proved for all j ≥ 0. For the second step, we prove that the sequence (U j ) j≥0 is a Cauchy sequence in the Banach space C([0, T 1 ]; H N−1 (R 3 )), which converges to the solution U = (ρ, u, c 1 , c 2 ) of the Cauchy problem (2.2)-(2.3), and satisfies sup 0≤t≤T 1 [U j (t)] H N−1 ≤ M. See for example [16].
For simplicity, we denote δ f j+1 := f j+1 − f j . Subtracting the j-th equations from the (j + 1)-th equations, we have the following equations for δρ j+1 , δu j+1 , δc j+1 1 and δc j+1 1 : The estimate of δρ j+1 is as follows: The estimate of δu j+1 is We have a similar way to estimate δc j+1 1 and δc j+1 2 as follows: We combine the equations (3.10)-(3.13) to obtain By using Gronwall's inequality, we obtain sup 0≤t≤T 1 By taking T 1 > 0 sufficiently small we find that (U j ) j≥0 is a Cauchy sequence in the Banach space C([0, T 1 ]; H N−1 (R 3 )). Thus, we have the limit function Thus, as j → ∞ the limit exists such that For the third step, we show that U j+1 (t) 2 H N is continuous in time for each j ≥ 0. For simplicity, let us define the equivalent energy functional for any 0 ≤ s ≤ t ≤ T 1 . The time integral on the right-hand side from the above inequality is bounded by (3.9), and hence EU j+1 (t) is continuous in t for each j ≥ 0. Therefore, For the fourth step, we show that the Cauchy problem (2.2)-(2.3) admits at most one solu- Multiplyingρ to both sides of the first equation of (3.15) and integrating over R 3 , we have Using integration by parts and the Cauchy-Schwarz inequality, we have Next, we establish the energy estimates forũ. By multiplyingũ to both sides of the second equation of (3.15) and integrating in x, we have By using integration by parts and the Cauchy-Schwarz inequality, we have Since L ∞ norms of ρ i , u i , c 1,i , c 2,i where i = 1, 2 are bounded, we have We have a similar way to estimatec 1 andc 2 as follows: By taking a linear combination of all estimates, we obtain The Gronwall's inequality implies sup 0≤t≤T 1 Since the initial data of (ρ,ũ,c 1 ,c 2 ) are all zero for T > 0, that implies the uniqueness of the local solution.

A priori estimates
In this subsection, we provide some estimates for the solutions for any t > 0. We use the energy method to obtain uniform-in-time a priori estimates for smooth solutions to Cauchy problems (2.2)-(2.3).
Proof. First, we find the zero-order estimates. For the estimate of ρ, multiplying ρ to both sides of the first equation of (2.2) and taking integrations in x ∈ R 3 , we obtain Using integration by parts and the Cauchy-Schwarz inequality, we have Now, we estimate u by multiplying the second equation of (2.2) by u and integrating over R 3 . Then, we have By using integration by parts and the Cauchy-Schwarz inequality, we have For the estimates of c 1 , we multiply c 1 to both sides of the equation of c 1 and integrate with respect to x, and we have By using integration by parts, we have Similar to above, from the equation of c 2 , we have Now, we make estimates on the high-order derivatives of (ρ, u, c 1 , c 2 ). Take α with 1 ≤ |α| ≤ N.
Applying ∂ α to the first equation of (2.2), multiplying by ∂ α ρ and then integrating in x, we have By using integration by parts and Cauchy-Schwarz inequality, we obtain Similarly for ∂ α u, what follows from (2.2) 2 is By using integration by parts and the Cauchy-Schwarz inequality, we have Similarly, we estimate c 1 , c 2 as follows: Then, after taking the summation over 1 ≤ |α| N and the combination (3.29) × d 1 + (3.30) + (3.31) + (3.32), we obtain  On the other hand, from a priori estimates, we have which is a contradiction to (3.34). Therefore, T = ∞ holds. This implies that the local solution U(t) obtained in Lemma 3.1 can be extended to infinity in time. Thus, we have a global solution (ρ, u, c 1 , c 2 )(t) ∈ C([0, ∞); H N ). This completes the proof of Proposition 2.1.

Linearized homogeneous system
In this section, to study the time-decay property of solutions to the nonlinear system (2.2), we have to consider the following Cauchy problem arising from the system (2.2)-(2.3) Here, the nonlinear source term takes the form To obtain the time-decay rates of the solution to the system (4.1) in the next section, we are concerned with the following Cauchy problem for the linearized homogenous system corresponding to (4.1) In this section, we always denote U 1 = [ρ, u] as the solution to the linearized homogeneous system with the initial data U 1 | t=0 = U 1,0 = (ρ 0 , u 0 ) in R 3 .
Similarly, by taking the Fourier transform for the second equation of (4.6), we get with initial dataû| t=0 =û 0 . Further, by taking the dot product of (4.8) withξ, we havẽ Here and in the sequel we setξ = ξ |ξ| for |ξ| = 0. Then, we have ρ t + in ∞ ξ ·û + n ∞ρ = 0 (4.10) We can rewrite (4.10) as withÛ(ξ, t) = (ρ(ξ, t),ξ ·û(ξ, t)) T and where T denotes the transpose of a row vector. Then, The eigenvalues of the system are as follows Therefore, the eigenvectors corresponding to the eigenvalues λ of A(ξ) that satisfy (A − λI) From the work above, one can define the general solution of (4.10) as From this, we deduce that It is straightforward to obtain Moreover, by taking the curl for the second equation of (4.6), we have Taking the Fourier transform in x for the above equation, we have Initial data is given as (ξ ×û)| t=0 =ξ ×û 0 . For t ≥ 0 and ξ ∈ R 3 with |ξ| = 0, one has the decompositionû =ξξ ·û −ξ × (ξ ×û). It is straightforward to get After summarizing the above computations on the explicit representation of the Fourier transform of the solution U 1 = [ρ, u], we have ρ(ξ, t) u(ξ, t) =Ĝ(ξ, t) ρ(ξ, 0) u(ξ, 0) .
We can verify the exact expression of the Fourier transformĜ(ξ, t) of Green's function G(ξ, t)= e tB aŝ

L 2 -L q time-decay property
In this subsection, we use (4.23) to obtain the refined L 2 -L q time-decay property for where e tB is the linear solution operator for t ≥ 0. For this, we need to find the time-frequency pointwise estimate onρ,û in the following lemma.

Time-decay rates for the nonlinear system
In this section, we will prove (2.5)-(2.7) in Proposition 2.2. The main idea is to introduce a general approach to combine the energy estimates and spectral analysis. We will apply the linear L 2 − L q time-decay property of the linearized homogeneous system (4.4), studied in the previous section, to the nonlinear case. We need the mild form of the original nonlinear Cauchy problem (2.2). Throughout this section, we suppose that U = [ρ, u, c 1 , c 2 ] is the solution to the Cauchy problem (2.3) with initial data U 0 = (ρ 0 , u 0 , c 1,0 , c 2,0 ). Then, by Duhamel's principle, the solution U = [ρ, u, c 1 , c 2 ] can be formally written as where e Bt U 0 is the solution to the Cauchy problem (4.1) with initial data U 0 = (ρ 0 , u 0 , c 1,0 , c 2,0 ). Here, the nonlinear source term takes the form (4.3).

Time rate for the energy functional and high-order energy functional
In this subsection, we will prove the time-decay rate for the energy functional U(t) 2 H N and the time-decay rate for the high-order energy functional ∇U(t) 2 H N . For that, we investigate the time-decay rates of solutions in Proposition 2.1 under extra conditions on the given initial data U 0 = [ρ 0 , u 0 , c 1,0 , c 2,0 ]. We define for an integer N ≥ 4. We also define E N U(t) ∼ [ρ, u, c 1 , c 2 ] 2 H N as the energy functional and H N as the dissipation rates. First, we start with this proposition for the energy functional and the high-order energy functional.
for t ≥ 0. Now, we proceed by making the time-weighted estimate and iteration for the inequality (5.5). Let l ≥ 0. Multiplying (5.5) by (1 + t) l and integrating over [0, t] gives Using (5.5) again, we have By iterating the above estimates for 1 < l < 2, we have To estimate the integral term on the right-hand side of (5.6), let us define E N,∞ (U(t)) = sup 0≤s≤T (1 + t) 3 Now, we estimate the integral term on the right-hand side of (5.6) by applying the linear estimate on u in (4.38) to the mild form (5.1), giving us Recall the definitions (4.3) of g 1 and g 2 . It is direct to check that for any 0 ≤ s ≤ t, Putting the above inequalities into (5.7), gives Next, we prove the uniform-in-time boundedness of E N,∞ U(t) which yields the time-decay rates of the energy functional E N U(t). In fact, by taking l = 3 2 + in (5.6) where > 0 is sufficiently small, it follows that Here, using (5.10) and the fact that E N,∞ (U(t)) is non-decreasing in t, it further holds that t 0 (1 + s) Therefore, it follows that and thus Since N+1 (U 0 ) > 0 is sufficiently small, it holds that E N,∞ U(t)) ≤ C 2 N+1 (U 0 ) for any t ≥ 0, which gives U(s) H N ≤ C(E N U(t)) 3 4 . This proves (5.3). Now, we estimate the high-order energy functional. By comparing the definitions of E N U(t), D N U(t) and D h N U(t), it follows from (5.5) that we have for any t ≥ 0. Similarly to obtaining (5.8), we estimate the time integral term on the (r.h.s.) of the above inequality. One can apply the linear estimate (4.29) to the mild form (5.1) so that Recall the definition (4.3) of g 1 and g 2 . It is straightforward to check that for any 0 ≤ s ≤ t, Putting this into (5.10) gives Then, by using (5.11) in (5.9), we have which implies (5.4). The proof of Proposition 5.1 is complete.

Conclusion
We have studied a chemotaxis model where a compressible fluid model for cells and a diffusive Lotka-Volterra model for chemoattractants and repellents are used. The previous results for chemotaxis are mostly extensions of the Keller and Segel model or in the case of fluid dynamical models, the incompressible fluid models for the cells are used. We showed the existence of global solutions and their asymptotic behavior in three dimensions with the initial data as a small perturbation of the constant state (n ∞ , 0, 0, 0). Our method is based on the basic energy estimates used for the a priori estimates and the iterative method in solving the Cauchy problem (1.1). Moreover, we have also shown the decay estimates of solutions to the Cauchy problem (1.1) in R 3 , in which the detailed analysis of Green's functions of the linear system is combined with the refined energy estimates with the help of Duhamel's principle. We proved the decay property of solutions as time goes to infinity. Our results are complementary to Ambrosi, Bussolino and Preziosi [2], where the modeling aspects such as qualitative analysis and numerical simulations of the compressible fluid model for cells with chemoattractants are examined for vasculogenesis.