Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation

This paper deals with a boundary-value problem for a coupled chemotaxis-Stokes system with logistic source \begin{document}$\begin{eqnarray*}\left\{\begin{array}{llll}n_t+u·\nabla n = \nabla·(D(n)\nabla n)-\nabla·(n \mathcal{S}(x, n, c)·\nabla c)\\ +ξ n-μ n^{2}, x'>in three-dimensional smoothly bounded domains, where the parameters $ξ\ge0$, $μ>0$ and $φ∈ W^{1, ∞}(Ω)$, $D$ is a given function satisfying $D(n)\ge C_{D}n^{m-1}$ for all $n>0$ with $m>0$ and $C_{D}>0$. $\mathcal{S}$ is a given function with values in $\mathbb{R}^{3×3}$ which fulfills \begin{document}$ \begin{equation*}{\label{1.3}}\begin{split}|\mathcal{S}(x, n, c)|\leq C_{\mathcal{S}}(1+n)^{-α}\end{split}\end{equation*}$ \end{document} with some $C_{\mathcal{S}}>0$ and $α>0$. It is proved that for all reasonably regular initial data, global weak solutions exist whenever $m+2α>\frac{6}{5}$. This extends a recent result by Liu el at. (J. Diff. Eqns, 261 (2016) 967-999) which asserts global existence of weak solutions under the constraints $m+α>\frac{6}{5}$ and $m\ge\frac{1}{3}$.


1.
Introduction. Chemotaxis is one particular mechanism responsible for some instances of such demeanor, where the organism like bacteria, adapts its movement according to the concentrations of a chemical signal. The chemotaxis-fluid Keller-Segel-Navier-Stokes model        n t + u · ∇n = ∆n − ∇ · (n∇c), x ∈ Ω, t > 0, c t + u · ∇c = ∆c − c + n, x ∈ Ω, t > 0, u t + κ(u · ∇u) = ∆u − ∇P + n∇φ, x ∈ Ω, t > 0, ∇ · u = 0, x ∈ Ω, t > 0 (1) arises in the modeling of bacterial populations, like Escherichia coli, in which the cells live in a viscous fluid so that cells and chemical substrates are transported with fluid, and that the motion of the fluid is under influence of gravitational forcing generated by aggregation of cells [40]. In addition, the model (1) is also introduced in [20,21] to study the coral broadcast spawning, where Ω ⊂ R N is a bounded domain with smooth boundary. In this setting, n = n(x, t), c = c(x, t), u = u(x, t) and P = P (x, t) denote the cell population density, the chemical concentration, the fluid velocity and the associated pressure, respectively. The coefficient κ is related to the strength of nonlinear fluid convection. If the fluid flow is relatively slow, we can use the Stokes equation instead of the Navier-Stokes equation. If all effects of fluid flow are ignored by letting u ≡ 0, model (1) can be reduced to quasilinear chemotaxis model n t = ∇ · (D(n)∇n) − ∇ · (nS(x, n, c)∇c), x ∈ Ω, t > 0, c t = ∆c − c + n, x ∈ Ω, t > 0 (2) which as an important variant of the classical chemotaxis Keller-Segel model [19] was proposed by Painter and Hillen [31] to model chemotaxis of cell populations. The signal is produced by the cells. The results about the chemotaxis model (2) appear to be rather complete, which concentrates on the problem (2) whether the solutions are global bounded or blow-up (see [1,4,5,7,8,29,11,12,13,14,15,17,18,10,30,31,41,50,51,52,54,55] and references therein for detailed results). Moreover, for system (2) with logistic term f (n) = n − µn 2 and D(n) ≡ 1, S(x, n, c) ≡ χ, it is known that an arbitrarily small µ > 0 can guarantee the boundedness of solutions in N = 2 [30], while for N > 2 solutions may blow up in finite time [10,50]. In particular, for N > 2, an appropriately large µ (as compared to the chemotactic coefficient χ) can exclude unbounded solutions [52,53]. It is known from these results that the main elements which determine the solution behavior are spatial dimension and the total mass of cells. In addition to considering the logistic dampening term, the nonlinear variants of chemotactic sensitivity S = S(n, c) ( [2,11,12]) and diffusivity ( [7,22,60]) have also been identified to prevent finite-time blow-up. From these works, it can be observed that the three different version of chemotaxis sensitivities: the signal-dependent sensitivity S(n, c) = C S c or C S (1+δc) 2 with C S > 0 and δ > 0 (see [9,54]) which reflects the inhibition of cell movement in the location of the high signal concentration [11,23], the n−dependent sensitivity S(n, c) = n q or n(n + 1) q with q ∈ R [14,36] which shows the volume-filling effect in the process of chemotaxis [31] and the tensor- [3,24] which describes the rotational chemotactic migration happening close to the physical boundary of the domain [61,62] have been deeply investigated by Cieślak and Stinner [5,6], Tao and Winkler [36,59] and Zheng et al. [64,66]. An important discovery is that the system (2) with tensor-valued sensitivity loses some energy structure as compared to that with scalar-valued sensitivities which leads to considerable difficulties in the mathematical analysis.
Recently, there have been increasing biological and mathematical interest in mathematical of the situation that the diffusion of bacteria (or, more generally, of cells) in a viscous fluid may be viewed like movement in a porous medium. Adjusting the above model accordingly, we shall subsequently consider the quasilinear Keller-Segel-Navier-Stokes system where n, c, u, P , φ and κ are mentioned as before. The function S measures the chemotactic sensitivity which may depend on n and D(n) is the diffusion function. The bacteria may proliferate following a logistic law with ξ ≥ 0 and µ > 0. If D(n) ≡ 1 in system (3) without logistic source, the global boundedness of classical solutions to the Stokes-version of system (3) with the tensor-valued S = S(x, n, c) satisfying |S(x, n, c)| ≤ C S (1+n) α with some C S > 0 and α > 0 which implies that the effect of chemotaxis is weakened when the cell density increases has been proved for any α > 0 in two dimensions [44] and for α > 1 2 in three dimensions [45]. If the signal-dependent functions S fulfilling S(n, c) ≤ C S (1+βc) ι with ι > 0 and β > 0, Liu el at. [27] proved that the two-dimensional Stokes-version of system (3) possesses a unique globally bounded classical solution and obtained a global weak solution for three-dimensional Navier-Stokes-version of system (3). For the system (3) with D(n) ≡ 1 and ξ, µ > 0, Tao and Winkler [35] proved that the three-dimensional Stokes-version of system (3) with S(x, n, c) ≡ C S > 0 and u−equation in (3) with a given external force g = g(x, t) possesses a globally classical solution for µ > 23 and the solutions of system (3) decay to zero for the case ξ = 0, while for any µ > 0 the analogous conclusion is obtained in the two-dimensional chemotaxis-Navier-Stokesversion of system (3) [34]. We further note that, with some exceptions such as [28,65], the result on global boundedness and large time behavior properties for the variant of (3) with nonlinear diffusion and nonlinear cross-diffusion is absent. More related results are obtained (see [25,26,39,43,47,68]).
It is noted that the second equation in system (1) describes the situation where the chemical signal is consumed and also secreted by cells, while if the signal is only consumed by the cells and the diffusion of bacteria in a viscous fluid may be viewed like movement in a porous medium, then system (1) is transformed into which was initially proposed by Tuval et al. [40] to describe the dynamics of the cell concentration, oxygen concentration and fluid velocity. To be more precise, they observed the large-scale convection patterns in a water drop through the fluidair interface. Here c denotes the oxygen concentration, f (c) is the consumption rate of the oxygen by the cells and φ is mentioned above. As to the mathematical analysis of system (4), numerous results on global existence and boundedness properties have been obtained for the variant of (4) obtained on assuming that ∇·(D(n)∇n) is linear diffusion ∆n [16,56,57,58]. Certain natural quasi-Lyapunov functional on the logarithmic entropy Ω n ln n guarantees the global solvability of system (4) under some suitable structural hypothesis on S and f [56,57] and also allows for the construction that the solutions of the two-dimensional Navier-Stokesversion stabilize to the spatially homogeneous equilibrium (n 0 , 0, 0) in the large time, where n 0 := 1 |Ω| Ω n 0 > 0 (see [58]). However, for the two-dimensional and three-dimensional (Navier-)Stokes-version system (4) with a tensor-valued sensitivity S as mentioned before, the energy-based reasoning does not guarantee the existence of global solution. Therefore, the method of the combinational functional Ω n p + Ω |∇c| 2 are developed by relaxing its excessive dependence on the inflexible structural assumptions on S and f such that f S ≤ 0 on (0, ∞) for the twodimensional system (4) with tensor-valued S [16]. If the bacteria diffuse in a porous medium (i.e. D(n) = mn m−1 ), S is the identity matrix I and f (c) = c, it is shown that the three-dimensional Navier-Stokes-version of system (4) possesses at least one global weak solution for m ≥ 2 3 [63] and that the three-dimensional chemotaxis-Stokes system (4) possesses at least one global weak solution [56]. As for the case of degenerate cell diffusion of porous medium type, Tao and Winkler [37,38] proved that the two-dimensional degenerate-chemotaxis-Stokes system possesses a bounded global weak solution for m > 1 and the three-dimensional degenerate-chemotaxis-Stokes has a locally bounded global weak solution for m > 8 7 . Besides that, Ishida [16] proved that the two-dimensional (Navier-)Stokes-version system (4) with ro- with some ϑ > 0 as well as D(n) ≥ C D n m−1 for all n ≥ 0 with m > 7 6 and C D > 0, and [48] developed an alternative a priori estimates to obtain that the three-dimensional chemotaxis-Stokes system (4) possesses at least one bounded weak solution which stabilizes to the spatially homogeneous equilibrium (n 0 , 0, 0) with n 0 as mentioned above as t → ∞. Very recently, it is shown [46] that the three-dimensional chemotaxis-Stokes system (4) with S(x, n, c) fulfilling with some C S > 0 and α > 0 possesses global weak solutions for m + α > 7 6 . More results about Cauchy problem are also obtained (see [32,67]).
As compared to the Navier-Stokes-chemotaxis (3) with the linear diffusion and logistic source, Liu and Wang [28] recently obtained global boundedness and decay property for the following three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation Due to the presence of the tensor-valued sensitivity, the corresponding chemotaxis-Stokes system (6) loses some energy structure, which gives rise to considerable mathematical difficulties. The authors [28] derived a prior estimates for ||n(·, t)|| L p (Ω) and ||∇v(·, t)|| L 2q (Ω) for all p, q > 1 by using an energy-like inequality. The main step is to estimate the two chemotaxis-related integral terms Ω n p+1−m−2α (·, t)|∇c(·, t)| 2 dx and Ω n 2 (·, t)|∇c(·, t)| 2q−2 dx for all m > 0 and α > 0. As compared to the methods in [28], we take a slightly different approach and apply a variant of the Gagliardo-Nirenberg inequality to control a part of the two chemotaxis-related integral term as mentioned above by the integral term Ω n p+1 (·, t)dx instead of the integral term Main results. Accordingly, the goal of the present work is mainly devoted to studying these questions for the three-dimensional Keller-Segel-Stokes system (6) with the tenor-valued sensitivity, and to give somewhat complete answer with regard to global existence, boundedness for general ξ, µ > 0. The core step is still to obtain the upper bound of the functional where n and c are components of the solutions to (17) below. In order to formulate our main results in this direction, let us specify the precise evolution problem addressed in the sequel by considering (6) along with the initial data and the boundary conditions D(n)∇n − nS(x, n, c) · ∇c · ν = 0, ∂c ∂ν = 0 and u = 0, x ∈ ∂Ω.
Furthermore, we suppose that the diffusion coefficient D fulfills as well as with m > 0 and C D > 0. As to the initial data, for simplicity we shall require throughout this paper that with A r standing for the Stokes operator with domain D( . Within the above framework, our main results concerning global existence and boundedness of solutions to (6) are stated as follows: Let Ω ⊂ R 3 be a bounded domain with smooth boundary, and let ξ ≥ 0 and µ > 0. Assume that D and S fulfill (10)- (11) and (5) with m + 2α > 6 5 . Then for any (n 0 , c 0 , u 0 ) satisfying (12), (6) admits at least one global weak solution (n, c, u, P ) in the sense of Definition 4.1 below. Also, this solution is bounded in Ω × (0, ∞) in the sense that ||n(·, t)|| L ∞ (Ω) + ||c(·, t)|| W 1,∞ (Ω) + ||u(·, t)|| W 1,∞ (Ω) ≤ C for all t > 0. (13) with some positive constants C. In addition, c and u are continuous in Ω × (0, ∞), and n as an L ∞ (Ω)−valued function is continuous on [0, ∞) with respect to the weak− * topology, i.e., Remark 1. We note that model (6) exists a global solution provided that m+2α > 6 5 , which implies that diffusivity and nonlinear variant of chemotactic sensitivity in (5) can prevent finite-time blow-up. Theorem 1.1 extends a recent result by Liu el at. [28] which asserts global existence of weak solutions under the constraints m + α > 6 5 and m ≥ 1 3 . The paper is organized as follows. After introducing the regularized system of (6) and collecting some basic estimates of the solutions in Section 2, we shall derive an upper bound for (7) by using a slightly different approach in Section 3. In Section 4, we construct the bounded global weak solution of (6) by passing to the limit in a standard manner on the basis of the previously established estimates.
2. Preliminary. In this section, we shall first deal with some boundary regularized approximate problem to overcome the difficulties brought by the nonlinear boundary condition in this section. As done in [24], we introduce an appropriate regularization in which S defined below vanishes near the lateral boundary.
Due to the presence of logistic source, some useful estimates for n , c and u can be derived from Lemma 2.2-Lemma 2.6 in [35].
Lemma 2.2. Let (n , c , u , P ) be the solution of (17). Then there exists a positive constant C > 0 independent of such that for all t ∈ (0, T max, ).
Proof. By Lemma 2.2-Lemma 2.6 in [35], we can find a positive constant C 1 such that thus there exists a constant C 2 > 0 such that Ω c 2 (·, t) ≤ C 2 for all t ∈ (0, T max, ), which is a direct consequence of the Poincaré inequality. This proves (20).
3. A priori estimates. In this section, we shall propose some regularity estimates for n and c by tracking the time evolution of a certain combinational functional of them. In order to establish some estimates for the coupled functional in (7), we first recall the following two lemmas proved in [28].
Next, in quite a similar manner, we can also estimate the first term on the right hand of (25).
Proof. By virtue of the Holder's inequality applied with exponents with p+1 2 and η = p+1 p−1 , we have Here we observe that the inequality p > 2 − m+2α 2 in particular ensures that and hence 2(q−1)(p+1) q(p−1) < 6. In view of the Gagliardo-Nirenberg inequality (see [49]) for a version and the boundedness of ∇c with respect to L 2 −norm, we can find some constants
Then for all η 3 > 0, the solution of (17) satisfies where a positive constant C 15 depends on q and η 3 .
With all above regularization properties of each component n , c , u at hand, we can obtain the following boundedness results by invoking a Moser-type iteration (see [37,Lemma A.1]) and standard parabolic regularity arguments.
Proof. In light of (23) with p > 3, we can infer from Lemma 2.3 with r := ∞ that Du is bounded in L ∞ (Ω × (0, T max, )) and thereby (53) is valid. Taking the results of Lemma 3.6 with properly large p and q as a starting point, we apply a Mosertype iteration to the n −equation in (17) and then get (51). Next we can apply the well-known arguments from parabolic regularity theory to the c −equation in (17) to obtain (52) on the basis of (40) and (42) (54).
In view of (19) and Lemma 3.7, the local-in-time solution can be extended to the global-in-time solution. Proposition 1 allows for an extension of the outcome in Lemma 3.7 from [0, T max, ) to [0, ∞). Next we will state the lemma.
Lemma 3.8. Let m + 2α > 6 5 . In addition, δ is supposed to be as in (12). Then one can find C > 0 independent of ∈ (0, 1) such that ||n (·, t)|| L ∞ (Ω) ≤ C for all t ∈ (0, ∞) and ||c (·, t)|| W 1,∞ (Ω) ≤ C for all t ∈ (0, ∞) as well as Moreover, we also have According to Lemma 3.18 and Lemma 3.19 in Winkler [48], we can also use the uniform bound (55)-(58) and the standard parabolic regularity theory to obtain the following uniform Hölder continuity for c , ∇c and u . Lemma 3.9. Let m + 2α > 6 5 . Then there exists ς ∈ (0, 1) such that for some C > 0 we have as well as and such that for each τ > 0 we can find C(τ ) > 0 such that To derive strong compactness properties, we shall need an appropriate boundedness property of the time derivative of certain power n . On time intervals of a fixed finite length, this can be achieved by making use of the priori bounds derived above.