BOUNDEDNESS AND LARGE TIME BEHAVIOR IN A TWO-DIMENSIONAL KELLER-SEGEL-NAVIER-STOKES SYSTEM WITH SIGNAL-DEPENDENT DIFFUSION AND SENSITIVITY

. This paper is concerned with the following Keller-Segel-Navier-Stokes system with signal-dependent diﬀusion and sensitivity ( ∗ ) in a bounded smooth domain Ω ⊂ R 2 with homogeneous Neumann boundary conditions, where a ≥ 0 and b > 0 are constants, and the functions d ( c ) and χ ( c ) satisfy the ) The diﬃculty in analysis of system ( ∗ is the possible of diﬀusion due to the condition c we will d ( function and the of to global existence


(Communicated by José A. Carrillo)
Abstract. This paper is concerned with the following Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity nt + u · ∇n = ∇ · (d(c)∇n) − ∇ · (χ(c)n∇c) + an − bn 2 , x ∈ Ω, t > 0, ct + u · ∇c = ∆c + n − c, x ∈ Ω, t > 0, ut + u · ∇u = ∆u − ∇P + n∇φ, x ∈ Ω, t > 0, ∇ · u = 0 x ∈ Ω, t > 0, ( * ) in a bounded smooth domain Ω ⊂ R 2 with homogeneous Neumann boundary conditions, where a ≥ 0 and b > 0 are constants, and the functions d(c) and χ(c) satisfy the following assumptions: • 1. Introduction and main results. Chemotaxis describes the movement of cells reacting to the presence of a chemical substance, which is called attractive/repulsive if the chemotactic movement is toward the higher/lower concentration of chemical substance. Chemotaxis process play an important role in embryonic development, tissue homeostasis, wound healing, as well as finding food and forming the multicellular body of protozoa [12]. The first mathematical model describing chemotaxis at population level was proposed by Keller and Segel [25], which was reduced to the following system under the quasilinear-steady-state assumptions for the chemical reaction n t = ∇ · (D(n, c)∇n) − ∇ · (S(n, c)∇c), c t = ∆c + n − c, where n = n(x, t) denotes the cell density and c = c(x, t) stands for the concentration of chemical substance. D(n, c) > 0 is a measure of the vigor of the random motion of the individual cell and S(n, c) > 0 is a measure of the strength of the influence of chemical concentration. The theoretical studies of system (1) have been done for different D(n, c) and S(n, c). And the most extensively studied case is D(n, c) = D 0 and S(n, c) = χ 0 n with constants D 0 > 0 and χ 0 > 0: which is called the classical Keller-Segel chemotaxis model [6] and whose solution behaviors strongly depend on the space dimensions (No blow-up in 1-D [19,33], critical mass blow-up in 2-D [17,23,31,32,36] and, generic blow-up in ≥ 3-D [45,47], see also the review articles [1,20,21]). Since the finite time blow-up of solutions limits the applicability of model to real world. Based on the biological inspiration or mathematical motivation, different mechanisms have been proposed to prevent finite time blow-up and exhibit the spatial pattern formations [18]. Among these, the cell self-interaction was considered as the important role, which affect both cell random motion and chemotactic movement [34] and can be described as where the diffusion coefficient φ(n) and chemotactic sensitivity function ψ(n) both depend on the cell density. For the non-flux initial-boundary value problem of system (2) in smoothly bounded domain Ω ⊂ R N (N ≥ 2), it has been proved that the blow-up or boundedness of solution depends on the values of θ, which is the ratio of φ(n) ψ(n) ≈ n θ at large values of n (see the review papers [1,20,21] and recent progress in [7,8,22,37,44,50] for details).
Since the chemical is secreted by the cells themselves, when cells get crowded during chemotactic movement, the chemical concentration also increases. Hence it is nature to consider the effects of the local interactions between cells and chemicals such as where the diffusion function depends on the chemical concentration c. However, to our knowledge, few mathematical results are available for the cell-chemical interaction chemotaxis system (3) except for the special case χ(c) = −d (c) > 0. In fact, if χ(c) = −d (c) > 0, the first equation of system (3) can be rewritten as n t = ∆(d(c)n), which together with the second equation was proposed in [14] to describe the stripe pattern driven by the density-suppressed motility. In this case, Yoon and Kim [51] considered a particular form of d(c) = c0 c k , c 0 > 0, k > 0 and showed the existence of global classical solution with uniform-in-time bound for any dimensions if c 0 > 0 is small. Tao and Winkler [42] recently established the existence of global classical solution in two dimensions and global weak solutions in three dimensions by assuming that d(c) has a positive lower and upper bound (i.e. 0 < d 1 ≤ d(c) ≤ d 2 for all c ≥ 0, where d 1 , d 2 are two constants). Furthermore, if the first equation of system (3) was replaced by n t = ∆(d(c)n) + µn(1 − n) (i.e., cell has growth), Jin et al. [24] studied the global dynamics and patterns ( like aggregation and wavefronts) without small assumption as imposed in [51] and the lower-upper bound assumption in [42].
In the above cell self-interaction chemotaxis model (2) or cell-chemical interaction chemotaxis model (3), the relevant influence of the cells or chemical on the surrounding habitat was ignored. However, some experiment evidence and numerical simulation indicated that liquid environment play an important role on the chemotactic motion of cells [5,30,43]. In this paper, we will study the following cell-chemical interaction chemotaxis model in the fluid environment where Ω ⊂ R 2 is a bounded domain with smooth boundary, u = u(x, t) and P = P (x, t) represent the velocity and the associated pressure of the fluid flow, respectively. Suppose that the given potential function φ = φ(x) satisfies where A := −P∆ denotes the realization of the Stokes operator in L 2 σ (Ω), defined on its domain D(A) = W 2,2 (Ω) ∩ W 1,2 0 (Ω) ∩ L 2 σ (Ω) with L 2 σ := {ψ ∈ L 2 (Ω)|∇ · ψ = 0 in D (Ω)}, and with P denoting the Helmholtz projection of L 2 (Ω) onto its closed subspace L 2 σ (Ω). Before stating our main results, we first recall some relevant results: If d(c) = d 0 and χ(c) = χ 0 are positive constants, Espejo and Suzuki [13] proved the global existence of certain weak solutions of system (4) in two dimensions with a = 0. After that, Tao and Winkler [40] investigated the global existence and large time behavior of classical solutions of the system (4) with a given external force g = g(x, t) in the Navier-Stokes fluid equation, and the results have been extended to three dimensions with Stokes fluid and the condition b ≥ 23 [41]. On the other hand, if the zero-term n − c is replaced by the absorption term −nf (c) in the second equation of system (4)(see [43] for details), Lorz [29] established the local existence of solution with a = 0 and b = 0. Furthermore, the global existence and large time behavior of solution were studied extensively with linear diffusion and porous medium diffusion (e.g. see [3,4,9,10,11,28,38,48,46,49] and the review article [1] for details). However, if the diffusion function d(c) depends on the chemical concentration c, to our knowledge, no much rigorous results for the system (4) have been available to date. The purpose of this paper is to establish the boundedness and large time behavior of solutions to system (4). These are not so trivial due to the following two difficult points. The first difficulty is the possibility of degeneracy for diffusion since d(c) → 0 as c → ∞, which may cause many difficulties to establish the global dynamics of solutions. The second difficulty comes from the fluid described by Navier-Stokes equation. To overcome this problems, we will use d(c) as a weight function and some techniques developed recently in [40] to obtain the global existence of classical solutions. Moreover, the large time behavior of solutions will be established by constructing Lyapunov functional. Our main results are stated as follows.
Let Ω be a bounded domain in R 2 with smooth boundary and d(c), χ(c) satisfy the hypotheses (H1) and (H2). Assume φ satisfies (5), and suppose the initial data n 0 , c 0 , u 0 satisfy (6). Then the problem (4) has a unique nonnegative global classical solution where , then the solution of (4) satisfies Remark 1. The results in Theorem 1.1 are new even in the fluid-free system (4) with u = 0. Indeed, in our results, we do not require the special structure χ(c) = −d (c), which plays an important role in [24,42,51].

Remark 2.
In fact, with the boundedness results obtained in Theorem 1.1, using the similar arguments as in [40], we can further obtain the following results on the large time behavior of solutions: If a = 0, then for any b > 0 the solution of (4) satisfies 2. Local existence and preliminaries. In what follows, without confusion, we shall abbreviate Ω f dx as Ω f for simplicity. Moreover, we shall use c i for C i (i = 1, 2, 3, · · · ) to denote a generic constant which may vary in the context. We first give the existence of local solutions of (4) by Schauder fixed point and the standard parabolic regularity theory.

Lemma 2.1 (Local existence).
Let Ω be a bounded domain in R 2 with smooth boundary and d(c), χ(c) satisfy the hypotheses (H1) and (H2). Assume φ satisfies (5), and suppose the initial data n 0 , c 0 , u 0 satisfy (6). Then there exists T max > 0 such that the problem (4) has a unique classical solution (n, c, u, P ) such that n ≥ 0 and c ≥ 0 inΩ × (0, T max ), and and T ∈ (0, 1) to be specified below, and in the Banach space we consider the closed convex set LetZ := (ñ,c,ũ) and Z := (n, c, u), then we introduce a mapping Φ : and u being the solution of Applying the variation of constants to (13), we obtain where (e tA ) t≥0 and P denote the Stokes semigroup with Dirichlet boundary data and the Helmholtz projection in L 2 (Ω), respectively. Choosing α ∈ ( 1 2 , 1), applying A α to both sides of (14) and using the regularization estimate for the Stokes 3600 HAI-YANG JIN semigroup [15], one has On the other hand, using the boundedness of ∇φ L ∞ and ñ(·, s) L 2 , one has Substituting (16) and (17) into (15), we have and hence Similarly, we apply the variation-of-constant formula to (12) to obtain Then using the semigroup estimates and (19), from (20) one has which gives c(·, t) W 1,∞ ≤ C(T ) for all t ∈ (0, T ). Then noting the fact d (c) < 0 in (H1) and c(·, t) L ∞ ≤ C(T ), we derived(c) has positive lower bound, which implies the equation in (11) is uniformly parabolic. Then noting the fact χ(c)∇c ∈ L ∞ (Ω × (0, T )) and a − bñ ∈ L ∞ (Ω × (0, T )) and using the parabolic regularity estimate ([27, Theorem V1.1]), one has n(·, t) for some θ ∈ (0, 1) and c 9 (R) > 0. From (22), we have and thus n(·, t) L ∞ ≤ n 0 L ∞ + c 9 (R)T Then we prove that Z ∈ S T and hence Φ maps S T into itself. Moreover, one can show the map Φ is compact in X. Using the Schauder fixed point, we can conclude that there exists a Z ∈ S T such that Φ(Z) = Z, the uniqueness can be proved by using the similar arguments as in [24,46]. (ii) Regularity and non-negativity.
Using the standard bootstrap arguments involving the regularity theories for parabolic equations and the stokes semigroup, we obtain the smooth properties listed in (10). The solution may be prolonged in the interval [0, T max ) with either T max = ∞ or T max < ∞ in the case At last, the non-negativity of solutions n and c follows from the classical maximum principle.
Next, we present some basic boundedness properties based on the integration in the first equation of system (4).
With the estimates in Lemmas 2.2 in hand, we can derive some basic timeindependent of c and u by using the similar argument as in [40,]. Lemma 2.3. There exists a constant C 3 > 0 such that and ∇c(·, t) L 2 ≤ C 3 for all t ∈ (0, T max ) (30) as well as where τ and T are defined by (26).
Proof. The estimates in (29)-(31) can be proved by using the similar argument as in [40, Lemmas 3.4-3.7], we omit the details of proof for convenience.
Next, we show some basic properties which will be used later.

Lemma 2.4 ([26]).
Let Ω be a bounded domain in R n with smooth boundary. Assume there is a constant C 4 > 0 such that g(·, t) L p ≤ C 4 , for all t ∈ (0, T ).
If z 0 ∈ W 1,∞ (Ω), then there exists a constant C 5 > 0 such that for every t ∈ (0, T ) the solution of the problem if p > n.
3. Proof of Theorem 1.1. In this section, we are devoted to proving Theorem 1.1.

3.1.
Boundedness. First, we establish the global boundedness of solutions. Due to the possible degeneracy ( lim c→∞ d(c) = 0), we can not directly used the diffusive dissipation as in [40] to obtain the boundedness of n(·, t) L p (p > 1) (see [40,Lemma 3.8]). To overcome the difficult, we will use the ideas as in [24] and employ the L 2 energy estimate directly by using d(c) as a weight function to pick up the advantage of diffusive dissipation to absorb the cross-diffusion and get the following Gronwall type inequality: which will lead to the uniform-in-time bound of n(·, t) L 2 by using the boundedness of Proof. We multiply the first equation of system (4) by n and integrate the result by parts, and then use the Hölder inequality and the Young's inequality to obtain On the other hand, using the fact Substituting (38) into (37), one has The assumptions in (H1) and (H2) imply that there exist two constants Using (40) and the Hölder inequality, we derive from (39) that Applying the Gagliardo-Nirenberg inequality and using the fact d(c) ≤ d(0) = c 2 , one has Ω |d 1 2 (c)n| 4 On the other hand, using Lemma 2.5 and noting the fact ∇c L 2 ≤ c 5 (see (30)), we obtain Ω |∇c| 4 Combining (42) and (43), and using the Young's inequality, one derive
In this subsection, we will show the boundedness of n(·, t) L ∞ . Before that, we first improve the regularity of u based on the ideas in [40,Lemma 3.11].
Proof. Let α ∈ ( 1 2 , 1) and the Stokes operator A defined on the domain D(A) := which is finite by the regularity properties. Hence to prove (48), we need to show M (T ) is finite for all T > 0. For given t ∈ (τ, T max ), letting t 0 := max{τ, t − 1} and applying the variation-of-constants formula associated with the third equation of system (4), we obtain Then, we apply A α to both sides of (50) and use the well known smoothing estimate [15] A α e −tA ψ L 2 ≤ c 1 t −α ψ L 2 for all ψ ∈ L 2 σ (Ω) and t > 0,
Next, we show the boundedness of c(·, t) L ∞ to exclude the possibility of degeneracy. Lemma 3.3. Suppose the conditions in Lemma 3.1 hold. Then there exists a con- and Proof. Noting the boundedness of u(·, t) L 2 in (29) and the estimate (58), then using the Gagliardo-Nirenberg inequality we obtain Let g(·, t) := n(·, t) − u(·, t) · ∇c(·, t), then the combination of (30), (36) and (61) gives Then applying Lemma 2.4 to the second equation of system (4) and noting (62), we have which implies (59) by Sobolev inequality. (60) follows directly due to d (c) < 0. Then the proof of Lemma 3.3 is completed.
where C 10 > 0 is a constant independent of t.
Lemma 3.5. Suppose that the conditions in Lemma 3.1 hold. Then the solution of system (4) satisfies where the constant C 11 > 0 independent of t.
Proof. Multiplying the first equation of system (4) by n p−1 with p ≥ 2, then we have Using the assumption (H1) and Lemma 3.3, one has |χ(c)| ≤ c 1 and d(c) ≥ d(K * ). Furthermore, nothing ∇c L ∞ ≤ c 2 and applying the Cauchy-Schwarz inequality,
Then we obtain the following results on the existence of global classical solution.
Lemma 3.6 (Boundedness of solution). Let the conditions in Theorem 1.1 hold.
Then the system (4) has a unique global classical solution satisfying (7).

3.2.
Large time behavior. In this subsection, we will study the large time behavior of solution for system (4). Suppose that We shall show that if b > K0 16 , the constant stationary state ( a b , a b , 0) is globally asymptotically stable. Motivated by the idea in [24,39], we will construct a Lyapunov functional to study the convergence of solutions. Next, we will show the convergence of u as follows.
Lemma 3.9. Let the conditions in Lemma 3.10 hold. Then it holds that u(·, t) L ∞ → 0 as t → ∞.
Proof. Multiplying the third equation of (4) by u, and using the fact u| ∂Ω = 0 and ∇ · u = 0, we obtain On the other hand, we use the Poincaré inequality and the fact u| ∂Ω = 0 to find a constant c 1 > 0 such that which, together with the boundedness of φ and the Young's inequality, gives Substituting (97) into (95), and using (96) again we obtain where c 3 = 1 c1 . Using (81) and the definition of F(t), we can find a constant c 4 > 0 such that Let y(t) := Ω |u| 2 and h(t) := 2c 2 Ω ∇n| 2 , then from (98) one has y (t) + c 3 y(t) ≤ h(t).
Combining (101) and (61), one has (94). Then we finish the proof of Lemma 3.9 In summary, we obtain the following results on the large time behavior. Then the solution of (4) satisfies n(·, t) − a b L ∞ → 0, c(·, t) − a b L ∞ → 0 and u(·, t) L ∞ → 0 as t → ∞.
Proof. Lemma 3.10 is a consequence of Lemma 3.8 and Lemma 3.9.
Proof of Theorem 1.1. Combining Lemma 3.6 and Lemma 2.1, we obtain the existence of global classical solution satisfying (7). Furthermore, the large time behavior of solution is proved in Lemma 3.10. Hence we finish the proof of Theorem 1.1.