Optimal time-decay rates of the Keller–Segel system coupled to compressible Navier–Stokes equation in three dimensions

Recently, Hattori–Lagha established the global existence and asymptotic behavior of the solutions for a three-dimensional compressible chemotaxis system with chemoattractant and repellent (Hattori and Lagha in Discrete Contin. Dyn. Syst. 41(11):5141–5164, 2021). Motivated by Hattori–Lagha’s work, we further investigated the optimal time-decay rates of strong solutions with small perturbation to the three-dimensional Keller–Segel system coupled to the compressible Navier–Stokes equations, which models for the motion of swimming bacteria in a compressible viscous ﬂuid. First, we reformulate the system into a perturbation form. Then we establish a prior estimates of solutions and prove the existence of the global-in-time solutions based on the local existence of unique solutions. Finally, we will establish the optimal time-decay rates of the nonhomogeneous system by the decomposition technique of both low and high frequencies of solutions as in (Wang and Wen in Sci. China Math., 2020, https://doi.org/10.1007/s11425-020-1779-7). Moreover, the decay rate is optimal since it agrees with the solutions of the linearized system.


Introduction
As described in the pioneering literature Keller-Segel [26], chemotaxis as a biological process is responsible for some instances of such demeanor, which is the directed movement of living cells (e.g., bacteria) that move towards a chemically more favorable environment under the effects of chemical gradients. We shall note that when no chemicals are present, the movement of cells is completely random. When an attractant chemical is present, the motility changes, and the tumbles become less frequent so that the cells move towards the chemical attractant. It is important for microorganism to find food (e.g., glucose) by swimming toward the highest concentration of food molecules. A lot of relevant mathematical models have been developed; see [3,14,15] for examples. Furthermore, in [8], it can be observed experimentally that bacteria are suspended in the fluid, which is influenced by the gravitational forcing generated by the aggregation of cells. Moreover, oxygen plays an important role in the reproduction of aerobic bacteria. For instance, bacillus subtilis often live in the thin fluid layers near the solid-air-water contact line in which the swimming bacteria move towards a higher concentration of oxygen according to the mechanism of chemotaxis. Further, we also note that oxygen concentration, chemical attractant, and bacteria density are transported by the fluid and diffuse through the fluid [6,8,29,38].
Concerning the chemotaxis models based on fluid dynamics, i.e., the chemotaxis-fluid system, there are two approaches: incompressible and compressible. For the incompressible case, Chae-Kang-Lee [4] and Duan-Lorz-Markowich [9] 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 [10] showed the global existence of an attraction-repulsion chemotaxis-fluid system with a logistic source. Tan-Zhou [35,36] proved the global existence and time-decay estimate of solutions to the chemotaxis-fluid system in R 3 with small initial data. Later Tan-Zhong-Wu further obtained the time-decay estimates of time-periodic strong solutions [34]. The interested readers are referred to [27,28,37,39,[43][44][45][46][47][49][50][51][52] for more mathematical results concerning the well-posedness and regularity of solutions of the various types of the chemotaxis-fluid system. For the compressible case, Ambrosi-Bussolino-Preziosi [1] discussed vasculogenesis using the compressible fluid dynamics for the cells and the diffusion equation for the attractant. Modeling aspects of vasculogenesis are studied in [2,12,33]. Recently Hattori-Lagha established the global existence and the temporal decay of the solutions for a three-dimensional compressible chemotaxis system with chemoattractant and repellent [13]. We mention that the temporal decay of solutions is hydrodynamic equations are hot topics; see [11,[16][17][18][19][20][21][22][23][24]42] and the references cited therein.
Motivated by Hattori-Lagha's temporal decay results in [13], we further investigated the optimal time-decay rates of strong solutions with small perturbation to the threedimensional Keller-Segel system coupled to the compressible Navier-Stokes equations, which models for the motion of swimming bacteria in a compressible viscous fluid in R 3 and reads as follows: (1.1) Next we shall introduce the notations in the above system of equations, which is called a (single) compressible chemotaxis-fluid system or compressible Keller-Segel-Navier-Stokes system.
The unknown functions ρ = ρ(t, x) and u = u(t, x) denote the density and velocity of fluids, resp. The unknown functions n = n(t, x) and c = c(t, x) represent density of amoebae and oxygen concentration, resp. λ 1 > 0 is the coefficient of shear viscosity, and λ 2 := ν + λ 1 /3 with ν being the positive bulk viscosity. φ = φ(t, x) is a given potential function. The smooth function P(·) > 0 is the pressure of fluid (depending on ρ), S(n) a given sensitivity parameter function, and the consumption rate of oxygen f (c) a step function [40], please refer to [52] for the different mathematical expressions of S(n) and f (c) corresponding to different environments. However, for the sake of simplicity, we assume that in this paper, For the investigation of the Cauchy problem of the system (1.1), we should pose the initial condition: In this paper, we prove the global existence of small perturbation solutions around some rest state for the Cauchy problem (1.1)-(1.2) and provide time-decay rates for the strong solutions. Moreover, the decay rate is optimal since it agrees with the solutions of the linearized system. It should be noted that Hattori-Lagha only gave the time-decay rates for the zero-order derivative of solutions in [13]. However, the novelty of this paper is that we further provide time-decay rates for all-order derivatives of solutions using a decomposition technique of both low and high frequencies of solutions as in [41].

Notations
Before stating our main result, we shall introduce some notations, which are used frequently throughout the paper. The notation C i > 0 (i ∈ Z + ) represents a fixed constant. For simplicity, we use the expression m n to mean m ≤ Cn, where C is a positive constant and varies from line to line.
For m ≥ 0 and p ≥ 1, the norms of Sobolev spaces H m (R 3 ) and W m,p (R 3 ) are denoted by · H m and · W m,p , resp. In particular, we will switch to use · L 2 and · L p for m = 0, resp. In addition, f (ξ ) is the Fourier transform of f (x) with respect to the variables x ∈ R 3 , that is f (ξ ) = F(f )(ξ ). We further define where m is a pseudo-differential operator. Let χ 0 (ξ ) and χ 1 (ξ ) be two smooth cut-off functions satisfying 0 ≤ χ 0 (ξ ), χ 1 (ξ ) ≤ 1 (ξ ∈ R 3 ) and for r 0 and R 0 satisfying Let χ 0 (D x ) and χ 1 (D x ) be quasi-differential operators of χ 0 (ξ ) and χ 1 (ξ ) resp., then for any given function f (x) ∈ L 2 (R 3 ), we can define its frequency distribution (f l (x), f m (x), f h (x)) as follows . Notice that f (x) can be expressed as follows

Main results
Now we state the main result of this paper.
for some constants ρ ∞ > 0 and n ∞ > 0. There exists a constant ε > 0 such that if

3)
and then the Cauchy problem of (1.1)-(1.2) with initial data admits a unique global-in-time solution (ρ, u, n, c), which satisfies (

1.4)
Furthermore, if the initial data (ρ 0ρ ∞ , u 0 , n 0n ∞ , c 0 ) is bounded in L 1 (R 3 ), then there exists a constant C > 0, such that, for any t ≥ 0, Now we shall introduce our main idea for deriving the optimal time-decay rates in (1.5). The main difficulty focuses on obtaining the energy estimates, which include only the highest-order spatial derivative of the solution ∇ 2 (ρ -1, u), which is essentially caused by the "degenerate" dissipative structure of the hyperbolic parabolic system. To get the dissipative estimate for ∇ 2 ρ, the usual energy method is to construct the interaction energy functional between u and ∇ρ using the pressure term in linearized momentum equations; see (3.27). It implies that both the first and second orders of the spatial derivatives of the velocity and the density should be involved in the Lyapunov functional Consequently, the L 2 -norms of the highest order and the first-order derivative of solutions have the same time-decay rate.
One of the main goals of this paper is to develop a way to capture the optimal time-decay rates for the highest order derivative of the solution to the Cauchy problem (1.1)-(1.2) if the initial perturbation is bounded in L 1 (R 3 ). Firstly, using the standard energy method, we establish estimate (3.24) of the energy functional D h (t) in (3.22). Secondly, motivated by the decomposition technique of both the low and high frequencies of solutions in [41], to get rid of the obstacle from the term R 3 ∇u · ∇ρ dx, we shall remove the low-mediumfrequency part of the term from D h (t) in (4.12), which requires a new estimate for the low-medium-frequency term (see Lemma 4.1 for detailed derivation).
The rest of this paper is organized as follows. In Sect. 2, for the convenience of analysis, we write the original system (1.1) as a perturbation form (2.3). In Sect. 3, we establish a prior estimates of solutions, and provide the global unique solvability for the Cauchy problem of (1.1)-(1.2). Finally, in Sect. 4, we will derive the optimal time decay rate for the non-homogeneous system (2.3) by the decomposition technique of both low and high frequencies of solutions as in [41].

Reformation of motion equations
To facilitate the proof of Theorem 1.1, we shall first reformulate the Cauchy problem (1.1)-(1.2). Obviously, we have then the following inhomogeneous system of equations is equivalent to (1.1): where we have defined that From now on, we renew to define βu by u, then the system (2.3) is reformulated as where

Global existence and uniqueness for the nonlinear system
In this section, we will prove the global well-posedness result in Theorem 1.1, that is, the global existence and uniqueness for the solutions of the chemotaxis-fluid system.

Unique solvability
First of all, we define a work space for the Cauchy problem of (2.6) and (2.7) as follows for any 0 ≤ T ≤ +∞. Then, we further introduce the results of local existence and a priori estimates of solutions in sequence.
Then, there exists a constant T 0 > 0 depending on σ 0 , u 0 , N 0 , c 0 H 2 (R 3 ) such that the Cauchy problem (2.6) and (2.7) has a unique solution (σ , u, N, c) ∈ (0, T 0 ), which satisfies Proof We can easily prove the above conclusion using an iterative method, the fixed point theorem, and the maxima principle. Interested readers can refer to [5,31] for the proof.
Proof The proof of Proposition 3.2 will be given in Sect. 3.2.
Remark 3.1 Here C 1 is independent of ε and δ, and δ = max {2ε, 3 In addition, by (3.1), we have Thanks to Propositions 3.1 and 3.2, we immediately get the global existence of unique solutions of the Cauchy problem (2.6)-(2.7) using a standard continuity argument.

Proof of Proposition 3.2
In this section, we aim to complete the proof of Proposition 3.2. The key step is to derive the energy estimates of the lower and higher derivatives of the solution (σ , u, N, c) of the Cauchy problem (2.6)-(2.7).

Lemma 3.1 Let the functional D l (t) be defined as follows
where α 1 and γ are two given constants.
resp., then summing the resulting identities up, and finally, using Young inequality and the integral by parts, we get Multiplying ∇(2.6) 1 , (2.6) 2 by u and ∇σ , resp., and then integrating by parts, we have For any given constant α 1 > 0, we use Young's inequality to get Adding 0≤k≤1 (3.5) to (3.6) and then using the above two inequalities, we get Next, we estimate the nonlinear part on the right-hand side of (3.7). By exploiting the Hölder inequality, Young inequality, Lemmas A.4-A.5, assumption (3.1), and integral by parts, we can get Recalling the definition of h i (i = 1, 2, 3) and then using the Hölder inequality, Young inequality, assumption (3.1), and integral by parts, we know that (3.11) Similarly, we have For the last two nonlinear terms in (3.7), we can use the integral by parts, Young inequality, Hölder inequality, assumption (3.1), and Lemmas A.4-A.5 to estimate that Now we proceed to estimate for c. Multiplying ∇ k (2.6) 4 by ∇ k c in R 3 , and then we integrate by parts to get It is easy to estimate that where α 1 is a fixed parameter that satisfies the following definition and γ := n 2 ∞ . This completes the proof of Lemma 3.1.
Next, we focus on the energy estimate of the highest derivatives of solutions.

Lemma 3.2 Let the functional D h (t) be defined as follows
Then we have
Proof Multiplying ∇ 2 (2.6) 1 -∇ 2 (2.6) 3 by ∇ 2 σ , ∇ 2 u and ∇ 2 N in L 2 (R 3 ), resp., then summing the resulting identities up, and finally, using the Young inequality and integral by parts, we can get Multiplying ∇ 2 (2.6) 1 , ∇(2.6) 2 by ∇u and ∇ 2 σ , resp., and integrating by parts, we have Let α 2 be a fixed constant. Addition of α 2 × (3.25) and (3.24) yields Next, we estimate the nonlinear term on the right-hand side of formula (3.26). By the Hölder inequality, Young inequality, assumption (3.1), Lemmas A.3-A.4, and the integral by parts, we have (3.27) Exploiting the integral by parts, we have Making use of Lemmas A.3-A.5, assumption (3.1), and the definition of h 1 , we can derive from the above inequality that where we have used the fact that Similarly, it is easy to estimate that Using Lemma A.5, Young inequality, assumption (3.1), and Hölder inequality, we obtain and Substituting the above result into (3.26), we have Multiplying ∇ 2 (2.6) 4 by ∇ 2 c in L 2 (R 3 ), we find that It is also easy to estimate that Let α 2 be a constant satisfying By the smallness of δ, we immediately get Lemma 3.2.
With Lemmas 3.1-3.2 in hand, we easily further obtain Proposition 3.2. In fact, keeping in mind the Young inequality and the definitions of D l , D h , we have where C 2 > 0 is a constant. Integrating the two inequalities in the above two lemmas over [0, t], thus (3.2) holds for the small enough δ. This completes the proof of Proposition 3.2.

Decay rates
In this section, we shall derive the decay-in-time rates for the Cauchy problem (2.6)-(2.7).
The proof will be broken up into three subsections. First, in Sect

Cancellation of a low-frequency part
Inspired by the observation of canceling the low-frequency part of solutions, we have the following conclusion.

Lemma 4.1 It holds that
where the positive constants C are independent of δ.
Proof Multiplying ∇(2.6) 2 by ∇∇σ L in L 2 , we integrate by parts and use (2.6) 1 to get Then, thanks to the Young inequality, we have By virtue of the Plancherel theorem and Lemma A.3, we have Adding α 2 × (4.3) and (3.23) together and using (4.4) and (A.1), we estimate that In addition, using frequency decomposition and adding μ 1 to both sides of (4.5), we can get , 1}, and then using the smallness of δ, we get In view of frequency decomposition, one gets It follows from the Young inequality and integral by parts that where we have used the fact 0 < α 2 < 1 4 . Thanks to (4.7) and (4.10), we can deduce that for a suitable constant C 3 , Consequently, by the Gronwall inequality, we conclude that This completes the proof of Lemma 4.1.

Decay estimates of the low-medium-frequency parts
Based on the temporal decay estimates from Fourier analysis of linearized systems, we can derive the estimates of the low-medium frequency part of solutions of the Cauchy problem. Next, we divide the derivation into three steps.
Step 1: we decouple the velocity u.
In fact, the estimate of u translates into the estimate of b and the estimate of pu.
Applying the Fourier transform to (4.13), we get that (4.15) We rewrite the above equations in a vector form: where (4.17) Define g = |ξ |, then the characteristic polynomial of matrix H is given as follows: where a 0 = 1, a 1 = (2 + ν)g 2 + n ∞ , The four solutions of the equation P(λ) = 0 are Step 2: We shall analyze the asymptotic of the low-intermediate frequency. ( σ , b, N, c), there exists a constant C 4 , and the following inequality holds

Proposition 4.1 For a solution to
Proof We can derive from the homogeneous linear equation Since 2α 3 r 0 ≤ 1 2 , we have which implies that there is a positive constant C 4 , such that for any |ξ | ≤ r 0 , Consequently, we immediately get (4.20).

Lemma 4.2
For any given constants r and R with 0 < r < R, there exists a positive constant j such that e -tG(|ξ |) ≤ Ce (j t) for all r ≤ |ξ | ≤ R and t ∈ R + . (4.32) For the system (4.21), the inequality (4.32) yields where r and R are any given positive constants.
Proof It is easy to check that the eigenvalues of H(ξ ) have positive real parts for sufficiently small g. Next, we further extend this fact to the case that no condition is required for large g.
By the Routh-Hurwitz theorem, the roots of the function P(λ) have a positive real part if and only if the following determinants are positive: By calculation, we obtain H 3 = a 3 (a 1 a 2 -a 0 a 3 )-a 2 1 a 4 > 0. We mediately see the conclusions in Lemma 4.2 hold; please refer to Sect. 3.3 in [7] for details.
The linearized equations of (4.14) under Fourier transform take the following form: By a direct calculation, it follows from (4.37) that for all |ξ | ≥ 0, Finally, exploiting the Fourier analysis of linear systems, we can show the temporal decay estimates for the low-intermediate part of the Cauchy problem solution in L 2 t L 2 x -norm. Let H be a matrix of the differential operators, which enjoys the following form  Applying the Fourier transform to (4.41) with respect to the variables x and solving the ordinary equation with respect to t, we obtain where H(t) = e -tH (t ≥ 0) is the semigroup that generated by the linear operator H and H(t)f := F -1 (e -tH ξf (ξ )) with Then, we have the following decay estimate.

Decay rates for the nonlinear system
Next, we establish the time decay estimates of solutions to the nonlinear problem (2.6) and (2.7). Let us consider the nonhomogeneous problem: Putting (4.55) into (4.61) yields (4.62) Moreover, by the Frequency decomposition and (A.1), we get for 0 ≤ k ≤ 2 (4.63) Lemma A.1 ([41]) For any given integers q, q 0 , q 1 with q 0 ≤ q ≤ q 1 ≤ m, it holds that and Proof The above inequalities can be easily verified by the definition of the frequency distribution and using the Plancherel theorem.