Existence of Smooth Solutions to Coupled Chemotaxis-Fluid Equations

We consider a system coupling the parabolic-parabolic Keller-Segel equations to the in- compressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up criteria. For two dimensional Navier-Stokes-Keller-Segel equations, regular solutions constructed locally in time are, in reality, extended globally under some assumptions pertinent to experimental observation in [20] on the consumption rate and chemotactic sensitivity. We also show the existence of global weak solutions in spatially three dimensions with rather restrictive consumption rate and chemotactic sensitivity.


Introduction
In this paper, we consider mathematical models describing the dynamics of oxygen diffusion and consumption, chemotaxis, and viscous incompressible fluids in R d , with d = 2, 3. Bacteria or microorganisms often live in fluid, in which the biology of chemotaxis is intimately related to the surrounding physics. Such a model was proposed by Tuval et al. [20] to describe the dynamics of swimming bacteria, Bacillus subtilis. We consider the following equations in [20] and set Q T = (0, T ] × R d with d = 2, 3:      ∂ t n + u · ∇n − ∆n = −∇ · (χ(c)n∇c), ∂ t c + u · ∇c − ∆c = −k(c)n, where c(t, x) : Q T → R + , n(t, x) : Q T → R + , u(t, x) : Q T → R d and p(t, x) : Q T → R denote the oxygen concentration, cell concentration, fluid velocity, and scalar pressure, respectively. The nonnegative function k(c) denotes the oxygen consumption rate, and the nonnegative function χ(c) denotes chemotactic sensitivity. Initial data are given by (n 0 (x), c 0 (x), u 0 (x)).
To describe the fluid motions, we use Boussinesq approximation to denote the effect due to heavy bacteria. The time-independent function φ = φ(x) denotes the potential function produced by different physical mechanisms, e.g., the gravitational force or centrifugal force. Thus, φ(x) = ax d is one example of gravity force, and φ(x) = φ(|x|) is an example of centrifugal force. Experiments in [20] suggest that the functions k(c) and χ(c) are constants at large c and rapidly approach zero below some critical c * . Hence, in [20], these functions are approximated by step functions, e.g., k(c) = κ 1 θ(c − c * ) and χ(c) = κ 2 θ(c − c * ) for some positive constants κ 1 and κ 2 . Also in [2], numerical simulation of plumes was obtained for the same species of bacteria in [20] in two dimensions. Furthermore, they assumed that the functions χ(c) and k(c) are constant multiples of each other, i.e., χ(c) = µk(c).
The main goals of this paper are to show the local existence of smooth solutions in two and three dimensions with the general condition on the oxygen consumption rate and chemotactic sensitivity, and to demonstrate global existence of smooth solutions in two dimensions and weak solutions in three dimensions with appropriate assumptions of χ(c), k(c), φ and initial data. Here we mention the related works for the result in this paper. If we ignore the coupling of the fluids, we obtain the angiogenesis type system. The classical model to describe the motion of cells was suggested by Patlak [17] and Keller-Segel [11,12]. It consists of a system of the dynamics of cell density n = n(t, x) and the concentration of chemical attractant substance c = c(t, x) and is given as αc t = ∆c − τ c + n, n(x, 0) = n 0 (x), c(x, 0) = c 0 (x), (1.2) where χ is the sensitivity and τ − 1 2 represents the activation length. The system in (1.2) has been extensively studied by many authors(see [9,10,15,16,21] and references therein). For the chemical consumption model by the cell or bacteria, we refer to the following chemotaxis model motivated by angiogenesis. (1. 3) The global existence of the weak solution to the system in (1.3) was obtained by Corrias, Perthame and Zaag [3,4] with a small data assumption of n 0 L d 2 . The bacterial movement toward the concentration gradient model in the absence of the fluid, i.e., u = 0, was recently studied. When u ≡ 0, χ(c) ≡ χ and k(c) ≡ c in (1.1), it was shown in [19] that there exists a uniquely global bounded solution if 0 < χ ≤ 1 6(d + 1) c 0 L ∞ .
If the flow of the fluid is slow, then the Navier-Stokes equations can be simplified to the Stokes equations. For the case χ(c) ≡ χ, Lorz [14] showed the local existence of the solution for the Keller-Segel-Stokes system. In two dimensions, Duan, Lorz, and Markowich [6] showed the global existence of the weak solution to the Keller-Segel-Stokes equations with the small data assumptions on c 0 , φ and the assumptions on the functions such that In [13], Liu and Lorz showed the global existence of a weak solution to the two-dimensional Keller-Segel-Navier-Stokes equations with similar assumptions on k and χ to those in (1.4). The equation of n in (1.1) could have been replaced by a porous medium equation, i.e., ∆n is replaced by ∆n m and the following Keller-Segel-Stokes system has been considered in [7].
In [7], Francesco, Lorz and Markowich showed the global existence of the bounded solution to (1.5) when m ∈ ( 3 2 , 2]. In [13], Liu and Lorz proved the global existence of the weak solution when m > 4 3 in three dimensions. For the Keller-Segel-Navier-Stokes system (1.1), Duan, Lorz and Markowich [6] showed the global-in-time existence of the H 3 (R d )-solution, near constant states, to (1.1), i.e., if the initial data (n 0 − n ∞ , c 0 , u 0 ) H 3 is sufficiently small, then there exists a unique global solution.
As mentioned earlier, the aim of this paper is to obtain the local-in-time existence of the smooth solution in two and three dimensions and the global-in-time existence of the classical solution to (1.1) in two dimensions under the minimal assumptions on the consumption rate and chemotactic sensitivity. Now we are ready to state our main results. The first result in this article is the local existence in time of the smooth solutions to (1.1). Comparing with the result in [6], we show the local-in-time existence without smallness of the initial data.
Theorem 1 (Local existence) Let m ≥ 3 and d = 2, 3. Assume that χ, k ∈ C m (R + ) and , then there exists a unique classical solution (n, c, u) of (1.1) satisfying for any t < T Remark 1 For simplicity, we denote We remark that if T is the maximal time of existence with T < ∞ in Theorem 1, then Secondly, we obtain two blow-up criteria for the system (1.1) depending on dimensions.
Theorem 2 Suppose that χ, k, φ and the initial data (n 0 , c 0 , u 0 ) satisfy all the assumptions presented in Theorem 1. If T * , the maximal time existence in Theorem 1, is finite, then one of the following is true in each case of two or three dimensions, respectively: Remark 2 Theorem 2 can be interpreted as follows: then the local solution can persist beyond time T , i.e., (n, c, u) ∈ L ∞ (0, T + δ; X m ) ∩ L 2 (0, T + δ; X m+1 ) for some δ > 0.
The third main result is the global existence of the smooth solutions in the two-dimensional spatial domain R 2 . Motivated by experiments in [20] and [2], we assume that the oxygen consumption rate k(c) and chemotactic sensitivity χ(c) satisfy the following conditions: (A) There exists a constant µ such that sup |χ(c) − µk(c)| < ǫ for a sufficiently small ǫ > 0.
We remark that assumption (A) plays a crucial role in obtaining LlogL × H 1 × L 2 type estimates.
Theorem 3 (Global existence in two dimensions) Let d = 2. Suppose that χ, k, φ and the initial data (n 0 , c 0 , u 0 ) satisfy all the assumptions presented in Theorem 1. Assume further that χ and k satisfy the assumptions (A) and (B) and φ ≥ 0. Then the unique regular solution (n, c, u) exists globally in time and satisfies for any Remark 3 If we approximate Heaviside functions using the smooth functions, then the experiments in [20] satisfy the assumptions (A) and (B). Furthermore, the assumptions on φ are satisfied by gravitational and centrifugal forces. Also we note that 2D numerical studies were performed under the assumption χ(c) = µk(c) in [2].
Our final main theorem is on the global-in-time existence of weak solutions in three dimensions. The notion of a weak solution of (1.1) is detailed in section 4 (see Definition 5). For existence of global weak solution, we need similar restrictions on k(c) and χ(c) as in Theorem 3. More precisely, compared to (A), we impose a slightly stronger assumption, denoted by (AA), which is given as follows: (AA) There exists a constant µ such that χ(c) − µk(c) = 0.
We are ready to state our last main result.
The rest of this paper is organized as follows. In Section 2, we prove local-in-time existence of the smooth solution for the two and three dimensional chemotaxis system with incompressible Navier-Stokes equations and obtain some blow-up criteria for the solution. In Section 3, we show the global in time existence of the regular solution in two dimensions. In Section 4, we establish the existence of a weak solution in three dimensions.

Local existence and blow-up criterion 2.1 Local existence
We first consider the chemotaxis system coupled with the Navier-Stokes equations in two and three dimensions. We show the local existence of solutions (n, c, u) in H m−1 × H m × H m space with m ≥ 3. Proof of Theorem 1.
We construct the solution sequence (n j , c j , u j ) j≥0 by iteratively solving the Cauchy problems on the following linear equations We first set (n 0 (x, t), c 0 (x, t), u 0 (x, t)) = (n 0 (x), c 0 (x), u 0 (x)). Then, using the same initial data to solve the linear Stokes type equations and the linear parabolic equations, we obtain u 1 (x, t), c 1 (x, t) and n 1 (x, t), respectively. Similarly, we define (n j (x, t), c j (x, t), u j (x, t)) iteratively. For this, we presume that c j and n j are nonnegative and show the existence and the convergence of solutions in the adequate function spaces. We show the nonnegativity of c j and n j at the end of the proof.
To prove the conclusion, i.e., to obtain contraction in adequate function spaces, we show the uniform boundness of the sequence of functions under our construction via energy estimates.
• (Uniform boundedness) We here show that the iterative sequences (n j , c j , u j ) are in X m := H m−1 × H m × H m space for all j ≥ 0. Observing that we have the following energy estimates: (i) The estimate of n j+1 Thus, the L ∞ norm of c j+1 is uniformly bounded, which implies that χ(c j ) and k(c j ) are uniformly bounded for all j ≥ 0.
• (Contraction) The estimate of this part is similar to that of the previous one. For convenience, we denote δf j+1 := f j+1 − f j . Subtracting the j-th equations from the (j + 1)-th equations, we have the following equations for δn j+1 , δc j+1 and δu j+1 : (2.7) (i) The estimate of δn j+1 . Using the following standard commutator estimates , we have the following estimate: (ii) The estimate of δc j+1 .
where we used the Mean Value Theorem for the last term.
where similar standard commutator estimates are used as in the case of δn j+1 . Using Young's inequality, we have From the above inequality, we find that (n j , c j , u j ) is a Cauchy sequence in the Banach space C(0, T ; X m ) for some small T > 0, and thus we have the limit in the same space.
• (Uniqueness) To show the uniqueness of the above local-in-time solution, we assume that there exist two local-in-time solutions (c 1 (x, t), n 1 (x, t), u 1 (x, t)) and (c 2 (x, t), n 2 (x, t), u 2 (x, t)) of (1.1) with the same initial data over the time interval [0, T ], where T is any time before the maximal time of existence. Letc(x, t) (2.8) Multiplyingñ to both sides of the first equation of (2.8) and integrating over R d , we have Multiplyingc and −∆c to both sides of the second equation of (2.8), we have Multiplyingũ to both sides of the third equation of (2.8) and integrating over R d , we have Summing the above estimates, we obtain , and the initial data of (c,ñ,ũ) are all zero, (c,ñ,ũ) are all zero for T > 0. That implies the uniqueness of the local classical solution.
• (Nonnegativity) For completeness, we briefly show that n j and c j are nonnegative for all j. To use induction, we assume c j and n j are nonnegative. If we apply the maximum principle to the equation of c j+1 in (2.1), we find that c j+1 is nonnegative (k(c j )n j is nonnegative). Let us decompose n j+1 = n j+1 Recall that the weak derivative of n j+1 ) (see e.g. [18]). Now multiplying the negative part (n j+1 ) − on both sides of the first equation of (2.1) and integrating over [0, t] × R 2 , we have Using Gronwall's inequality, we have Since the initial data n j+1 0 is nonnegative, we conclude that n j+1 is nonnegative. This completes the proof.

Blow-up criterion
Next, we observe a blow-up criterion for the fluid chemotaxis equations.
Proposition 1 (A Blow-up criterion) Suppose that χ, k, φ and the initial data (n 0 , c 0 , u 0 ) satisfy all the assumptions presented in Theorem 1. If T < ∞ is the maximal time of existence, then Proof. At first, we consider the L 2 estimate of n. Multiplying n to both sides of the equation of n and integrating, we have Since χ is continuous and c is uniformly bounded until the maximal time of existence, we have For the estimates of c, we use the calculus inequality Multiplying −∆c to both sides of the equation of c and integrating, we obtain For the equations of u, multiplying −∆u to both sides of the equations and integrating by parts, we have Collecting all the estimates, we obtain Note that n L ∞ (0,T ;L 2 ) and ∇n L 2 (0,T ;L 2 ) are uniformly bounded if From the above inequality, we have n(t) L q ≤ C, where C is independent of q. Letting q → ∞, we have n ∈ L ∞ x L ∞ t . Next, we consider the estimate in the space (n, c) ∈ H 1 × H 2 . We have From Young's inequality and Gronwall's inequality, we have Then, we consider the estimate in the space (n, c, u) ∈ H 2 × H 3 × H 3 . Proceeding similarly to the above, we obtain In the above, the last term can be controlled by and Hence, if we use Young's inequality and Gronwall's inequality, we have Similarly, we estimate c as We can control the term (k(c)n) 2 . For the estimate of u, we have Let us consider H m−1 × H m × H m estimates. The case m = 2, 3 and 4 are proved in the above, hence we consider the m ≥ 5 case. Taking ∂ α (|α| ≤ m − 1) and multiplying ∂ α n to both sides of the equation n and integrating and summing, we have We already obtained the estimate for the case m = 4, thus ∇c L ∞ (0,T ;L ∞ ) is bounded. Hence, we have Using the classical product lemma on each step of iteration, we can control Then we have χ(c)n∇c H m−1 ≤ C(1 + c H m + ∇c m L ∞ ) n H m−1 using the product lemma. For the H m estimate of c, we proceed similarly to have As is shown for the term χ(c)n∇c H m−1 , we control the term (k(c)n) 2 For the estimate of u, we have Thus, by collecting all the above estimates and using Gronwall's inequality, we have (n, c, u) ∈ . This completes the proof. We are ready to present the proof of Theorem 2. Proof of Theorem 2.
In the proof of Proposition 1, we notice that ∇c L ∞ is solely responsible for n ∈ L 2 x L ∞ t and ∇n ∈ L 2 x L 2 t . Indeed, (2.10) Moreover, we have n ∈ L q x L ∞ t and ∇n q/2 ∈ L 2 x L 2 t for all 2 < q < ∞; Next, we see that ∇c ∈ L 2 We first consider the two-dimensional case.
L 2 , and use (2.9), (2.10). Similarly, we show that n ∈ L ∞ t H 1 Finally, we show that ω ∈ H 1 x L ∞ t ∩ H 2 x L 2 t . Testing −∆ω to the equations, we have d dt where 3 < p ≤ ∞ and 1/p + 1/l = 1/2. Note that, via the Gargliardo-Nirenberg's inequality, We treat the term u L p ∇ω L l ∇ 2 ω L 2 similarly to u L p ω L l ∇ω L 2 in the estimation of (2.13). Therefore, since ∇ 2 ω ∈ L 2 x L 2 t , we have This completes the proof.

Global solutions in two dimensions
In this section, we provide the proof of global existence of smooth solutions in time with large initial data in two dimensions. For the proof of Theorem 3, we show some a priori estimates, which are uniform until the maximal time of existence. Moreover, such estimates imply that the blow-up condition quantity in Theorem 2 is uniformly bounded up to the maximal time of existence. Therefore, the maximal time cannot be finite. Now we present the proof of Theorem 3.

Proof of Theorem 3.
We first present the following estimates for the solutions to the two-dimensional chemotaxis system coupled with the Navier-Stokes equations.
We have the mass conservation for n(t, x) as Multiplying c q−1 (t, x) to both sides of the second equation of (1.1) and integrating over R 2 , we have 1 q Hence, we have c ∈ L ∞ (0, T ; L q ) for any 1 < q ≤ ∞ and ∇c q 2 ∈ L 2 (0, T ; L 2 ) for any 1 < q < ∞. Multiplying ln n to both sides of the first equation of (1.1) and integrating over R 2 , we have Multiplying −∆c to both sides of (1.1) and integrating over R 2 , we obtain d dt Multiplying µ to both sides of (3.7) and then adding (3.6), we have where we used the condition (A). Here we choose ǫ to be so small that ǫC 3 < 2 and, for convenience, set λ 1 2 := C 2 c 0 2 L ∞ . On the other hand, multiplying u to both sides of the third equations of (1.1) and integrating over R 2 , we have Multiplying φ to both sides of the first equation of (1.1) and integrating over R 2 , we have Summing (3.9) and (3.10), we have d dt Multiplying λ 1 to both sides of (3.11) and adding (3.8), we obtain Using Gronwall's inequality, we have sup 0≤t≤T R 2 n ln n + µ|∇c| 2 + λ 1 2 |u| 2 + λ 1 nφdx Next, we show that n |ln n| ∈ L ∞ (0, T ; L 2 (R 2 )), following a typical argument for dealing with kinetic entropy (see e.g. [5]). We first note that n ln n ≤ C (3.14) This deduces the estimate (3.13). Next, integrating (3.12) in time t, we get R 2 n(·, t) ln n(·, t) + µ ∇c(t) 2 The term R 2 nu∇ x dx is bounded as follows: Noting that |∇ x | + |∆ x | ≤ C, we get where we used that n L 2 ≤ C n 0 In summary, we obtain where δ is sufficiently small, which will be specified later. Therefore, integrating (3.17) in time, Now adding 2 n(ln n) − to both sides of (3.15), we obtain R 2 n(·, t) |ln n(·, t)| + µ ∇c(t) 2 where δ in (3.17) is so small that term t 0 ∇ √ n 2 L 2 is absorbed to the left hand side of (3.15). Since (3.19) holds for all t until the maximal time of existence, due to Gronwall's inequality, we obtain n |ln n| ∈ L ∞ (0, T ; L 2 (R 2 )). Moreover, again via the inequality (3.19), we deduce (3.1)-(3.3).
We note that from the blow-up criterion in two dimensions in Theorem 2, it suffices to show that ∇c ∈ L 2 (0, T ; L ∞ (R 2 )) for global existence of smooth solutions in R 2 . We first consider the vorticity equation of velocity fields. Taking curl, we have where ∇ ⊥ = (−∂ 2 , ∂ 1 ). If we multiply ω to both sides of the above equation and integrate over R 2 , then we have Hence, we have ω 2 L ∞ (0,T ;L 2 ) + ∇ω 2 L 2 (0,T ;L 2 ) ≤ C n 2 L 2 (0,T ;L 2 ) . Since Next we consider the equation of n. Multiplying n and integrating over R 2 , we have where we used that χ is C 1 and c ∈ L ∞ (0, ∞; L ∞ ), i.e., χ(c) and χ ′ (c) are bounded. Due to Young's inequality, we have Therefore, via Gronwall's inequality, we have n ∈ L ∞ (0, T ; L 2 ) ∩ L 2 (0, T ; H 1 ). Multiplying ∆ 2 c to both sides of the equation of c and integrating over R 2 , we have We note that the last term above is controlled as follows: +C n L 2 ∇n L 2 ∇c L 2 ∇ 2 c L 2 + C ∇n 2 L 2 . Gronwall's inequality gives c ∈ L ∞ (0, T ; H 2 )∩ L 2 (0, T ; H 3 ), which implies via embedding that ∇c ∈ L 2 (0, T ; L ∞ ). This completes the proof.

Global weak solution in three dimensions
In this section we will show the global existence of the weak solutions for (1.1) in three dimensions. We start with notations. H 1 0 (R 3 ) is used to indicate the closure of compactly supported smooth functions in H 1 (R 3 ) and H −1 (R 3 ) means the dual space of H 1 0 (R 3 ). We also introduce the function spaces V(R 3 ), V σ (R 3 ), H(R 3 ) defined as follows: u ∈ L ∞ (0, T ; L 2 (R 3 )), ∇u ∈ L 2 (0, T ; L 2 (R 3 )).
(b) The functions n, c, and u solve the chemotaxis-fluid equations (1.1) in the sense of distributions, namely for any (c) The functions n, c and u satisfy the following energy inequality: with C = C(T, χ(c) L ∞ , x n 0 L 1 , ∇c 0 L 2 , n 0 | ln n 0 | L 1 , ∆φ L ∞ , ∇φ L ∞ , φ L ∞ ). Now we compute a priori estimate of an energy inequality under the Assumption (AA) and (B). We note first, by maximum principle, that It is straightforward that n(t) L 1 = n 0 L 1 for t ≥ 0 and d dt Multiplying µ to the last equation (4.3) and adding it to the second equation (4.2), we have d dt for some C 1 , which can be taken bigger than 1, i.e. C 1 > 1. Also it holds that Since the term R 3 nu∇ x dx is bounded as follows: We estimate the term R 3 χ(c)n∇c∇ x dx similarly as above.
Multiplying C 1 to (4.1) and adding it together with (4.4) and (4.6), we have d dt Then, by Gronwall's inequality, we have where C(T, χ(c) L ∞ , n 0 L 1 , x n 0 L 1 , ∆φ L ∞ , ∇φ L ∞ ). By same reasoning for treating n(ln n) − term in (3.14), it follows that Streamline of constructing global weak solutions, as in usual steps for the Navier-Stokes equations, is the following: · regularizing the system for which we prove the existence of smooth solutions · finding uniform estimates for the solutions of the regularized system · passing to the limit on the regularized parameters.

Regularization
In this subsection, we intend to construct approximate solutions of the system. For the incompressible Navier-Stokes equations defined on a general bounded domain, the global weak solutions are constructed by using the spectral projections (P k ) k∈Z , associated to the inhomogeneous Stokes operator ([1, Chapter 2]). A number of useful properties of the family (P k ) k∈Z are listed as follows: For any u ∈ H(Ω), In particular, (4.12) implies P k u ∈ L ∞ (Ω) for u ∈ L 2 (Ω) in three dimensions.

Definition 6
The bilinear map Q is defined by From now on we denote by H k (R 3 ) the space P k H(R 3 ). We regularize (1.1) by a frequency cut-off operator P k and a mollifier σ ǫ : ∂ t c k,ǫ (t) = −u k,ǫ · ∇c k,ǫ + ∆c k,ǫ − k(c k,ǫ )(n k,ǫ * σ ǫ ), ∂ t u k,ǫ (t) = −P k Q(u k,ǫ , u k,ǫ ) + P k ∆u k,ǫ − P k (n k,ǫ ∇φ), (4.14) with initial data (n k,ǫ 0 , c k,ǫ 0 , u k,ǫ 0 ) = (n 0 * σ ǫ , c 0 * σ ǫ , P k u 0 * σ ǫ ), where n 0 , c 0 , u 0 is the initial data of (1.1) satisfying the condition (1.8) in Theorem 4. The mollifier is defined as usual such that σ ǫ (x) = ǫ −3 σ(ǫ −1 x) for σ ∈ C ∞ 0 (R 3 ). Apart from the frequency cut-off the regularization is same one for a chemotaxis-fluid model studied in [13]. Repeating similar arguments in Theorem 1, we obtain the local solution of (1.1) in the class for some time T and for all m > 3. It turns out that due to the regularization of nonlinear terms and smoothing properties of P k (see (4.12)), the local solution of (1.1) can be extended up to infinite time.
Proposition 2 The regularized system (4.14) has the unique global solution (n k,ǫ , c k,ǫ , u k,ǫ ) in a class (4.15) for any time T < ∞.
Before presenting the proof we observe that the approximating solution (n k,ǫ , c k,ǫ , u k,ǫ ) of (4.14) satisfies an energy inequality.
Now we give the proof of Proposition 2.
Proof of Proposition 2 We first observe that the regularity criterion in Theorem 2 hold true for the system (4.14). Since its verification is tedious repetition of that of Theorem 2, we omit its details. If we consider the second equation of (4.14), then we have the following energy estimates.
By using Gronwall's inequality, we have ∇c k,ǫ we can also demonstrate that the Serrin condition in Theorem 2 is satisfied for u k,ǫ . This completes the proof.