Global existence and large time behavior for the chemotaxis–shallow water system in a bounded domain

In this paper, we consider the chemotaxis–shallow water system in a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^2 $\end{document} . By energy method, we establish the global existence of strong solution with small initial perturbation and obtain the exponential decaying rate of the solution. We divide the bounded domain into interior domain and the domain up to the boundary. In the interior domain, the problem is treated like the Cauchy problem. In the domain up to the boundary, the tangential and normal directions are treated differently. We use different method to get the estimates for the tangential and normal directions.


1.
Introduction. Chemotaxis-shallow water system was first proposed in [3] to describe the movement of bacteria cells under the influence of the signals of chemical and the fluid transportation. In this paper, we consider the chemotaxis-shallow water system          n t + div (nu) = D n ∆n − ∇ · (nχ(c)∇c), c t + div (cu) = D c ∆c − nf (c), h t + div (hu) = 0, hu t + hu · ∇u + h 2 ∇n + 1 2 (1 + n)∇h 2 = µ∆u + (µ + λ)∇( div u) in a bounded domain. Here, the unknown functions n, c, h, u represent density of bacterial, substrate concentration, the fluid height and the fluid velocity field respectively. Two constants D n and D c are diffusion coefficients of cells and substrate. Given functions χ(c) and f (c) denote the chemotactic sensitivity and the rate of substrate consumed by cells respectively. µ is the constant representing the shear viscosity and µ > 0. λ is the bulk viscosity coefficients satisfying µ + λ ≥ 0.
Here n is the density of cells and c is the concentration of chemical. It was proposed in [14] and [15]. The initial boundary value problem of the classical parabolicparabolic Keller-Segel model is x ∈ Ω, t > 0, c t = ∆c − c + n, x ∈ Ω, t > 0, ∂n ∂ν = ∂c ∂ν = 0, x ∈ ∂Ω, t > 0, n(x, 0) = n 0 , c(x, 0) = c 0 , x ∈ Ω, where Ω ⊂ R n is a bounded domain with smooth boundary. ∂ ∂ν := ν · ∇ and ν = (ν 1 , ν 2 ) is the outward unit normal vector on ∂Ω. For 1D case, the global solution exists and is bounded (see [20]). For 2D case, it was proved in [7] and [19] that the global solution exists and is bounded if n 0 L 1 < C * . If n 0 L 1 > C * , blowup of the solution was presented in [9], [10] and [24]. Here C * = 8π if the domain Ω is radially symmetric and C * = 4π if the domain Ω is non-radially symmetric. For higher dimensions, it was proved in [1] and [27] that the global solution exists and tends to a constant state with an exponential decaying rate. For the radially symmetric initial data (n 0 , c 0 ), blow-up of the solution in finite time was obtained in [29].
In the case Ω ⊂ R 2 , for k = 0, the global existence and steady state in L ∞ norm for initial boundary value problem were obtained in [28] and [30]. It was further proved in [34] that the solution tends to equilibrium with an exponential decaying rate. Later, under weaker conditions, the global existence was established in [5]. For k = 0, under rater general assumptions on χ, the global existence and asymptotic theories were achieved in [32] and [33]. In the case Ω ⊂ R 3 and k = 0, the global weak solution was obtained in [31] and the global stability was given in [2]. For the chemotaxis in a compressible fluid, as we know, only the chemotaxisshallow water system has been considered. There are many results for the Cauchy problem of this system. One could refer to [3] for the local existence of strong solution and a blow-up criterion. If the bacterial density does not allowed to vanish, the global existence of strong solution and the L p decay estimates for the global solution were studied in [22]. If the bacterial density is allowed to vanish, the global existence of classical solution and the L p decay estimates for the global solution were obtained in [25]. In order to overcome the difficulties of vacuum, in [25], we use the low and high frequency decomposition method. For the initial boundary value problem, only the local existence of strong solution and a blow-up criterion were obtained in [3].
The main purpose of this paper is to establish the global existence of strong solution and study the asymptotic behavior of the global solution in a bounded domain of R 2 . In the study of the initial boundary value problem, for the usual Keller-Segel system, we can solve the problem by energy estimates. But for this coupled-system, as the boundary conditions are different, it is difficult to get estimates for high order derivatives since it is always highly nontrivial to obtain boundary information for high order derivatives. It becomes the main trouble to apply energy method. However, it is also difficult to apply Green's function method or other spectral method to solve the initial boundary value problem. Recently, there are successful applications of Green's function in a bounded domain for chemotaxis model alone (without fluid equations) (see [27]). The difference is that chemotaxis model alone is a pure parabolic system and the Green's function in fact is just heat kernel while the coupled model is parabolic-hyperbolic and the corresponding Green's function is far more complicated. In fact, to the best of our knowledge, for the initial boundary value problem in a bounded domain of pure parabolic system (instead of parabolichyperbolic system), the Green's function method is well developed (see [16] and [27]). In this paper, motivated by [18] and [21], we divide the bounded domain into interior domain and the domain up to the boundary. For the previous one, we do not need to deal with the boundary information and it is like a Cauchy problem. To get the estimates of the latter one, we make full use of the structure of the system to convert a high order normal derivative into a high order tangential derivative.
Before we list the main result, we introduce some notations. Throughout this paper, ∂ t stands for the derivative with respect to time variable and f t = ∂ t f . The symbol ∂ i f (i = 1, 2) means partial derivative with respect to x i We employ the notation D l f to mean the partial derivative of order l. That is if l is a nonnegative integer, then is a set of all partial derivatives of order l, endowed with the norm where D α f := ∂ α1 1 ∂ α2 2 f. As usual, we use D to stand for D 1 . In the following, we define ξ = (τ, η) is a new coordinate transformed from the original coordinate x = (x 1 , x 2 ). The symbols ∂ l τ and ∂ l η stand for the tangential and normal derivatives of order l. Here l is a nonnegative integer. When l = 1, we omit the superscript l. For simplicity, we denote f dx = Ω f dx. The notationf = 1 |Ω| f dx. Now the main result can be stated as follows.

CHEMOTAXIS-SHALLOW WATER SYSTEM IN A BOUNDED DOMAIN 6383
The rest of this paper is organized as follows. In Section 2, we give the preliminary knowledge of this paper. In Section 3, we get the basic estimates. In Section 4, estimates are established in the interior domain. In Section 5, we investigate the estimates in the domain up to the boundary. In Section 6, we complete the proof of the main theorem.
2. The preliminary knowledge. In this section, we establish the local existence of strong solution and prove the nonnegativity of n and c. The proof of the local existence is analogous to the discussions in [3] and [4]. We only sketch the proof for completeness.
By the characteristic method, the existence and regularity of (6) 3 can be obtained. The solution can be represented as

WEIKE WANG AND YUCHENG WANG
The equation (6) 2 is a linear parabolic equation. Thus the existence, uniqueness and regularity of c can be achieved by the standard energy method for the linear parabolic equation. Then we can get the estimates for n by the same way. As the equation (6) 4 is also a linear parabolic equation, utilizing the estimates for (h, n) and the energy method, we obtain the existence, uniqueness and regularity of u.
The above argument is that if (ñ,ũ) ∈ M, then we can define a map mapping M to itself Φ : (ñ,ũ) → (n, u).
The proof of the following lemma can be found in [5]. 3. The basic estimates. In this section, we establish the basic estimates which can be obtained in the domain Ω itself without dividing it into interior domain and the domain up to the boundary.
where is suitably small.
Proof. Multiplying the system (8) byh 2 v, (nh) 2 c,n(1 +n)ρ, (nh)u respectively, integrating over Ω and summing up, according to the boundary conditions and integration by parts, we havē Applying Poincaré inequality and integrating (11) with respect to time variable, it is easy to obtain the inequality (10). where Proof. We use ∂ t to (8), multiply the resulting system byh 2 v t , (nh) 2 c t ,n(1 + n)ρ t , (nh)u t respectively, integrate over Ω and sum up to get One has Substituting (14) into (13) and then integrating the resulting equation with respect to time variable, we obtain the inequality (12).
Lemma 3.4. For Dv and Dc, we have and Proof. Multiplying (8) 1 , (8) 2 by v t , c t respectively and integrating over Ω, we can easily get the estimates.
Proof. Applying D l to equation (8) 3 , the construction of the resulting equation asserts Combining (15) and (15) integrating with respect to time variable, we derive Therefore, we complete the proof of this lemma.
For v t , c t , u t , we obtain the estimates for high order derivatives with respect to spatial variable Proof. Now we consider the system Multiplying the above system by −h 2 ∆v t , −(nh) 2 ∆c t , −(nh)∆u t respectively, integrating with respect to time and spatial variable gives the inequality (16).
In order to get the high order derivatives with respect to spatial variable, we need to divide the bounded domain into finite subdomains. Assume that {Ω j } n j=0 is a cover of Ω a.e. Ω ⊂ n j=0 Ω j and satisfies • Ω 0 ⊂ Ω and Ω j ∩ ∂Ω = ∅ (j = 1, · · · , n); • In each Ω ∩ Ω j (j = 1, · · · , n), it is given locally by a smooth function ψ j Here ψ j ∈ C 2 and (y 1 , y 2 ) is a new coordinate introduced in Section 5; is also a cover of Ω. Here • In each Ω j (j = 0, 1, · · · , n), we define a cut-off function χ j ∈ C ∞ 0 and In this paper, the domain Ω 0 is called the interior domain and {Ω j } n j=1 is called the domain up to the boundary. Below, we assume diam(Ω j ) (j = 1, · · · , n) is small such that δ ψ = max 1≤j≤n ψ j L ∞ (Ωj ) is suitably small. 4. The interior estimates. In this section, we aim to establish the estimates in the interior domain. In the interior domain, we do not need to consider the boundary conditions. We first estimate the high order derivatives with respect to ρ. We have Proof. First we consider l = 1. Applying D to (8) Multiplying (19), (8) 4 by µχ 2 0 Dρ andh 2 χ 2 0 Dρ respectively, integrating over Ω 0 and summing up, one gets µ 2 It follows from integration by parts that Now we turn to the nonlinear term In order to deal with χ 2 0 u t Dρdx, following the idea in [8], making use of equation (8) 3 , we can replace time derivative by the spatial derivative, that is

CHEMOTAXIS-SHALLOW WATER SYSTEM IN A BOUNDED DOMAIN 6389
Above all, we have Then we consider 2 ≤ l ≤ 3. Operating D l and D l−1 on the equations (8) 3 and (8) 4 respectively, it has that and Multiplying the equations (22), (23) by µχ 2 0 D l ρ andh 2 χ 2 0 D l ρ respectively and integrating over Ω 0 and summing up gives From (20), we deduce that each term of can be controlled by χ 0 D l ρ 2 L 2 and u 2 H l . Let us deal with χ 2 0 D l−1 u t D l ρdx as follows The nonlinear term can be estimated as

WEIKE WANG AND YUCHENG WANG
and Combining all the above estimates, yields the inequality (18).
Proof. First, we consider l = 1. Applying D to (8) and multiplying byh 2 χ 2 0 Dv, (nh) 2 χ 2 0 Dc,n(1 +n)χ 2 0 Dρ, (nh)χ 2 0 Du respectively, it gives that We use integration by parts to get For the nonlinear term χ 2 0 D(u∇ρ)Dρdx, it has been deal with in Lemma 4.1. Hence, it has that We notice that each term of R 1 can be controlled by In the following, we use R i to conclude these terms that the derivatives loss on the cut-off function. We do not concern about the specific form of R i . We know that it is easy to be estimated.
Then we turn to 2 ≤ l ≤ 3. Applying D l to (8) and multiplying byh 2 Repeating all the above procedures, we obtain inequality (24).

5.
The estimates in the domain up to the boundary. In this section, we establish the estimates in the domain up to the boundary. We use the different method to get estimates in the tangential and normal directions respectively. For any point x 0 ∈ ∂Ω, we can establish a new coordinate y = (y 1 , y 2 ) by shifting and rotating the coordinate x = (x 1 , x 2 ). The point x 0 is the origin of y-coordinate, the direction of y 1 -axis is parallel with the tangent of ∂Ω at the point x 0 and the positive direction of y 2 -axis is inward pointing into Ω. The coordinate (y 1 , y 2 ) can be expressed by (x 1 , x 2 ) is a translation vector. Then using τ = y 1 , η = y 2 − ψ j (y 1 ) to express the boundary Ω ∩ Ω j .
Through the new coordinate ξ = (τ, η), we transform the boundary of Ω ∩ Ω j into η = 0. So the tangential direction always satisfies the boundary conditions. For convenience, we only consider in one Ω j . We omit the subscripts j. According to the transformation, we have In the new ξ-coordinate, for simplicity, we introduce the following notations Then we can rewrite the equation (8) as Here In the following discussion, we mainly base on the ξ-coordinate.

5.1.
The estimates for tangential derivatives. The main purpose of this subsection is to get the high order tangential derivatives with respect to v, c, u.

WEIKE WANG AND YUCHENG WANG
and We observe that each term in R 3 can be easily estimated. In the following, we use R i to represent all these terms that the derivatives loss on the cut-off function and ψ. R i is not involved in our estimates and thus its expression is not needed. It follows from a priori assumption and Sobolev inequality that Together with (27) and (28), yields that Then we get the estimates for 2 ≤ l ≤ 3. Applying ∂ l τ to equation (25) and multiplying byh 2 ∂ l τ v, (nh) 2 ∂ l τ c,n(1 +n)∂ l τ ρ, (nh)∂ l τ u 1 , (nh)∂ l τ u 2 respectively and integrating, one gets Repeating the above procedures and combining all the estimates, we have (26).

5.2.
The estimates for normal derivatives. In this section, we will get the estimates for normal derivatives. As the equation of ρ does not have strongly elliptic operator ∆, we must get the high order normal derivatives with respect to ρ. Using ∂ η to (25) 3 and multiplying the resulting equation by (µ/h)(1 + (ψ ) 2 ), we have where dρ dt = ρ t + u∇ρ.
In the following, we give the Stokes lemma. It is very important because we can get some derivatives with respect to u and ρ at the same time by this lemma.
where g 0 ∈ H m+1 and g 1 ∈ H m . Then the system (37) has a solution (ρ, u) satisfying If we take ∂ k τ to (7) and use the cut-off function χ on the resulting equation, through Lemma 5.3, we have 6. The proof of Theorem 1.1. Now we combine the Lemmas 3.1-5.3 to close a priori assumption.
Lemma 6.1. For the system (2)-(4), we have Proof. First we get the estimates for low order derivatives.
I. By Lemma 3.2, we have the basic estimate II. By Lemma 3.5, for l = 0, we have the estimate L 2 )ds and for l = 1, we have III. By Lemma 3.3, we have the estimate IV. By Lemma 4.1, for l = 1, we have and for l = 2, we have V. By Lemma 4.2, for l = 1, we have and for l = 2, we have VI. By Lemma 5.1, for l = 1, we have and for l = 2, we have VII. By Lemma 5.2, for m = k = 0, we have for m = 1, k = 0, we have and for m = 0, k = 1, we have ( χD 2 ∂ τ u 2 L 2 + χD∂ τ ρ 2 L 2 )ds IX. By Lemma 3.4, we have . Now combining all the above inequalities, we have L 2 )ds.
Repeating the above procedures, we can get the estimates for high order derivatives.
Now we complete the proof of Theorem 1.1.