Global bounded solution of a 3D chemotaxis-Stokes system with slow $p$-Laplacian diffusion and logistic source

1.   Introduction

In 1970, Keller and Segel first proposed the classical chemotaxis model in ^[10], which explained the phenomenon of cellular slime mold aggregation in response to a chemical signal of increased concentration. The mathematical expression of the classical chemotaxis model with consumption is as follows:

After extensive research conducted by mathematicians, this model has produced excellent results, and relevant research results on the properties of solutions can be consulted in ^{[1,12,14,20,22,26,31]}.

Nevertheless, the interaction between chemicals and cells in their surroundings is evident from a variety of research and is unavoidable. Therefore, Tuval et al. ^[23] constructed the chemotaxis-fluid system with consumption in order to characterize that aerobic bacterial populations are suspended in sessile water droplets as follows:

The function $n$ indicates the density of bacteria and $c$ represents the concentration of oxygen, respectively. The fluid velocity field is denoted by $u$, and the rate at which bacteria consumes substrate is expressed by the function $f(c)$. $P$ is the associated pressure, and $\Phi$ is a function that represents potential, while the strength of nonlinear fluid convection is measured by $\kappa\in\mathbb{R}$. In 2010, Lorz ^[17] first obtained the local existence result in a bounded domain. In two-dimensional domains, according to ^[4], the classical solution exists globally with the small initial datum, and Winkler ^[29] established that classical solutions exist globally for the chemotaxis-Navier-Stokes model. Even some scholars have researched that the scalar chemotactic sensitivity function $\chi(c)$ is replaced by the matrix function $S(x, n, c)\in C^2(\bar{\Omega}\times [0, \infty)^2;\mathbb{R}^{3\times3})$ in the system (1.2), and we can make reference to ^{[3,9,15,24,32]}.

Consulting some chemotaxis literature ^[2,11], we know that the random movement of bacteria appears to be enhanced in close proximity to high concentrations, owing to the limited size of the bacteria. Therefore, it is natural to investigate nonlinear diffusion. Some researchers conclude that the solution is bounded when the diffusion parameters meet certain conditions for the 3D chemotaxis-fluid model with nonlinear diffusion in ^[5,25,27,28]. We discuss one of the forms of nonlinear diffusion known as $p$-Laplacian diffusion. Regarding the chemotaxis-fluid model with $p$-Laplacian diffusion, mathematicians have conducted research on various biological populations within three-dimensional space. Liu ^[16] studied a three-dimensional chemotaxis-Stokes model describing coral fertilization with arbitrarily slow $p$-Laplacian diffusion, and it is demonstrated that the global boundedness of solutions exists whenever $p > 2$. Han and Liu ^[6] investigated a 3D chemotaxis-Navier-Stokes system involving two species and $p$-Laplacian diffusion within smooth bounded domains and proved that if $p > 2$, the model admits a global weak solution. Tao and Li ^[19] changed the system (1.2) by setting $\chi(c) = \chi$ and replacing the $\Delta n$ term with $p$-Laplacian diffusion $\nabla\cdot(|\nabla n|^{p-2}\nabla n)$, investigated the subsequent model

in a three-dimensional bounded domain $\Omega$, and they proved that there exists the weak solution under the assumption that $p > \frac{32}{15}$. Subsequently, let $\chi = 1$ and $\kappa = 0$ in (1.3), and it becomes a chemotaxis-Stokes system; they improved the result presented in ^[18]. It indicates that for any $p > \frac{23}{11} (\thickapprox 2.09091)$, the system exists global bounded weak solutions. In a recent literature by Jin ^[8], the range of values has been extended to $p > p^*(\thickapprox 2.01247)$.

As we all know, the classical chemotaxis model (1.1), which contains the logistic source term that promotes the boundedness of solutions, can successfully inhibit the bacterial aggregation effect. Does the chemotaxis-fluid model retain this property? The answer is yes. In the three-dimensional domain, Lankeit ^[13] investigated the chemotaxis-Navier-Stokes model with a logistic growth term and demonstrated that the weak solution eventually becomes smooth and converges to a steady state after some waiting time. The 3D chemotaxis-Stokes model, which involves porous diffusion and the logistic source, was recently considered by Yang and Jin ^[30]. They demonstrated that under the large time limit, the solutions converge to the constant steady state and proved the boundedness of the weak solution with $m > 1, 0 < \alpha < 2m-1$. Captured by the papers above, a natural question struck us: Does a bounded weak solution exist for any slow $p$-Laplacian diffusion model with a logistic source? Hence, this work considers the chemotaxis-Stokes model

which includes slow $p$-Laplacian diffusion and a logistic growth term, inside a bounded domain $\Omega\subset\mathbb{R}^3$. The classical logistic term $\mu n(1-n)$ denotes the rate at which cells proliferate or die and, here, the parameter $\mu > 0$ and the function $\Phi\in W^{1, \infty}(\Omega)$.

Furthermore, it is assumed that every triple $(n_0, c_0, u_0)$ of initial data satisfies

Ahead of presenting the main result, it is necessary to give a concise definition for global weak solutions.

Definition 1.1. If $(n, c, u)\in X_{1}\times \ X_{2}\times X_{3}$, $\nabla\cdot u = 0$, such that for arbitrary testing functions $\phi_1, \phi_2 \in C^\infty_0(\bar{\Omega}\times [0, \infty))$, $\psi \in C^\infty_0(\bar{\Omega}\times [0, \infty); \mathbb{R}^3)$, and $\nabla\cdot\psi = 0$, the equalities

hold, then we call $(n, c, u)$ a global weak solution of the model (1.4), satisfying the initial conditions (1.5). Here,

Based on the Definition 1.1 and previous assumptions (1.5), our major result is given below.

Theorem 1.1. If $p > 2$, then the system (1.4) with the initial conditions (1.5) exists global weak solutions $(n, c, u)$ in the bounded domain $\Omega\subset\mathbb{R}^3$, such that for all $\beta\in(\frac{3}{4}, 1), s > 1$, satisfying

where positive constants $M_1, M_2$ only depend on $n_0, c_0, u_0, p, \Omega$. Here $Q(t) = \Omega\times(t, t+1)$.

The subsequent sections of this work are structured in the following approach. Section 2 will present a set of fundamental lemmas. Section 3 will provide an analysis of the energy estimate to demonstrate that the solution to the approximation problem exists globally and is bounded. Subsequently, we give the proof of Theorem 1.1 and the boundedness of global weak solutions by utilizing the approximation method.

2.   Preliminaries

Within the section, we present fundamental conclusions that will be referenced multiple times in the subsequent sections of the work.

Lemma 2.1. [7, Lemma 2.4] Suppose $a > 0, \delta\ge0, b\ge 0, T > 0$ and $\tau\in(0, T)$. Let $g: [0, T)\rightarrow [0, \infty)$ be an absolutely continuous function satisfying

where $h(t)\in L^1_{loc}([0, T))$ is a nonnegative function, and for all $t\in[\tau, T)$ fulfills

Then we have

Lemma 2.2. [7, Lemma 2.5] Let $\omega_0\in W^{2, q}(\Omega)$, $h\in L_{loc}^{q}((0, +\infty); L^q(\Omega))$, $\forall1 < q < \infty$, and one can find a fixed positive constant $\tau$ such that

Then the system

exists a unique solution $\omega\in L_{loc}^{q}((0, +\infty); W^{2, q}(\Omega)), \omega_t\in L_{loc}^{q}((0, +\infty); L^q(\Omega))$ such that

where the constant $M$ is independent of $\tau$.

Lemma 2.3. [30, Lemma 4.2] If $\Omega$ is a bounded domain and $\varphi\in C^2(\bar{\Omega})$ satisfying $\left.\frac{\partial\varphi}{\partial\nu}\right|_{\partial\Omega} = 0$, then there exists an upper bound on the curvatures of $\Omega$ given by $\kappa > 0$, such that

Lemma 2.4. 8, Lemma 2.3] If $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary and $\omega\in C^2(\bar{\Omega})$ satisfying $\left.\frac{\partial\omega}{\partial\nu}\right|_{\partial\Omega} = 0$, then the following two inequalities hold:

3.   Global existence and boundedness of the weak solution

With the goal to get the weak solution under the Definition 1.1 in three dimensions, we concerned the following regularized problem with $\varepsilon\in(0, 1)$:

where $n_{\varepsilon 0}, c_{\varepsilon 0}, u_{\varepsilon 0}\in C^{2+\alpha}(\bar{\Omega}), \Phi_\varepsilon\in C^{1+\alpha, \frac{\alpha}{2}}(\bar{\Omega}\times[0, +\infty))$ with

Firstly, to obtain our result, we recall the local existence result of the chemotaxis-Stokes model (1.4) as follows. The proof is similar to ^[18], thus we omit it.

Lemma 3.1. Let $p > 2$, then there exists a maximal time $T_{max}\in(0, +\infty)$ and a unique nonnegative solution $(n_\varepsilon, c_\varepsilon, u_\varepsilon)\in C^{2+\alpha, 1+\frac{\alpha}{2}}(\bar{\Omega}\times[0, T_{max}))$ of the model (3.1). Moreover, if $T_{max} < \infty$, then

Subsequently, we derive some estimates by using the maximum principle and straightforward calculations. These estimates are fundamental in the demonstration of our result.

Lemma 3.2. There exist nonnegative constants $C_1, C_2, and\; C_3$ such that the solution of (3.1) fulfills

where $C_1, C_3$ are independent of $\varepsilon, T_{max}, \tau$, while $C_2$ is independent of $\varepsilon$ and $T_{max}$.

Proof. By integrating the first equation in (3.1), we can derive

Shifting the terms gives

Then employing Lemma 2.1, we deduce

that is (3.2).

Then one can apply the maximum principle to $(3.1)_2$, and it can yield (3.3), which is the basic estimate of the solution component $c_\varepsilon$.

Regarding (3.4), we first multiply the second equation of (3.1) by $c_\varepsilon$ and make use of an integration by parts, which implies

collecting with (3.3), then we have

Furthermore, we test the third equation in (3.1) by $u_\varepsilon+ u_{\varepsilon t}$ and use the integration by parts to obtain that

Applying Poincaré's inequality and H$\ddot{\text{o}}$lder's inequality, we get

substituting (3.5) into the above inequality, then we gain

On the other hand, we first move the term of the third equation (3.1), which yields $-\Delta u_\varepsilon+\nabla P_\varepsilon = -u_{\varepsilon t}+n_\varepsilon\nabla \Phi_\varepsilon$, and use the $L^2$ theory of Stokes operator to obtain

By a combination of (3.5)–(3.7), we can derive (3.4).

Next, an energy inequality related to the variables $n_\varepsilon$ and $c_\varepsilon$ in the model (3.1) will be built.

Lemma 3.3. Let $p > 2$, then there exists a constant $C > 0$, which is independent of $\varepsilon, T_{max}$ and $\tau$ such that

Proof. The first equation of (3.1) is multiplied by $K(1+\ln{n_\varepsilon})$, where $K > 0$ is a constant to be determined, to obtain

which implies

By testing the second equation of (3.1) by $-\Delta c_\varepsilon$ and an application of Young's inequality, then one can find constants $C_2, C_3 > 0$ such that

Taking $K = 4C_2$ and $\eta = \frac{1}{2}$ in (3.10), then combining (3.4) and the above inequality, we arrive at

where $C_4$ is a positive constant. Finally, using (3.4), we can deduce (3.9).

Lemma 3.4. The triple $(n_\varepsilon, c_\varepsilon, u_\varepsilon)$ is the solution to the approximation system (3.1). If

where $C > 0$ is a constant, then for all $r > 2$

Proof. Taking advantage of Duhamel's principle, we can express $u_\varepsilon$ by

Then we have

Let $q = \frac{3}{2}$ in the above inequality, in view of (3.11), we can derive that

Lemma 3.5. If $p > 2$, then we can find some positive constant $C_r = C_r(r)$ such that

Proof. The first equation of (3.1) is multiplied by $n_\varepsilon^r$, then we integrate it by $\Omega$ to get

Combining the Young inequality and the assumption $p > 2$ ensures that $\frac{p}{p-1} < 2$, then we have

Shifting the above inequality, then using the Young inequality once again, we gain

Next, applying the Gagliardo-Nirenberg interpolation inequality to the above inequality, it yields

Meanwhile, combining Lemma 2.2 with (3.4), we arrive at

Taking $q = \frac{(r+2)(p-1)}{2p-3}$ in the above inequality, then substituting it into (3.13), we get

thanks to $\frac{p-1}{2p-3} < 1$. Next, let $\frac{(r+2)(p-1)}{2p-3} = \frac{5}{2}$, which implies $r = 3-\frac{5}{2(p-1)}$. Substituting it into (3.15) and shifting the terms, we arrive at

By employing H$\ddot{\text{o}}$lder's inequality and the Gagliardo-Nirenberg inequality, it becomes evident that

Combining Lemma 3.2, (3.16), and (3.17), we can deduce that

We notice that $4-\frac{5}{2(p-1)} > \frac{3}{2}$. Combining Lemma 3.4, for any $r > 0$, we have

Then collecting (3.4) and (3.18), we notice that

Taking $r = 4-\frac{3}{p-1}$ in (3.15), and combining the above inequality, we can deduce

We employ the gradient operator $\nabla$ on the first equation of (3.1) and test the resulting identity by $|\nabla c_\varepsilon|^{r-2}\nabla c_\varepsilon$, $\forall r > 2$, then a combination of Lemma 2.3 entails

Utilizing the boundary trace embedding inequality and (3.9), for any small $\eta_2 > 0$, we get

Thus, by employing inequality (2.2) and a combination of the above two inequalities, we can find a fixed positive constant $\sigma$ such that

Additionally, by applying the Gagliardo-Nirenberg inequality, we get

here $\beta = \frac{3(p-2)}{p-1}$, that is $2+\beta = 5-\frac{3}{p-1}$. Combining (3.9) with (3.19), we infer that

Next, multiplying the third equation of the system (3.1) by $u_{\varepsilon t}$, and integration by parts, we derive

Through a simple calculation, one can get

In accordance with (3.19) and (3.21), using the fact $5-\frac{3}{p-1} > 2$, we can arrive that

Moreover, we notice that

Taking $r = 4$ in (3.20) and collecting (3.21), the fact that $\frac{5p+\beta p+3\beta}{3} = \frac{8p^2-2p-18}{3p-3} > 3$ for any $p > 2$ and (3.22), we have

which implies

For any $r > 0, p > 2$, due to $\frac{2p}{p-1} < 4, r+\frac{1}{p-1}\le r-1+p, 2r+\frac{2}{p-1}\le\frac{3p}{(3-p)_+}\frac{p+r-1}{p}$, multiplying the first equation in the system (3.1) by $n_\varepsilon^r$, and employing the Sobolev embedding inequality and (3.23), we conduce

then by concise calculation, it derives (3.12) straightly.

Lemma 3.6. If $p > 2$, then one can find positive constants $M_1 = M_1(\beta), M_2, M_3 = M_3(r)$, such that

Proof. Noticing $5-\frac{3}{p-1} > 2$ for any $p > 2$ and (3.19), we can get

The Sobolev embedding theorem gives us the following inequality

where $\beta > \frac{3}{4}$. Combining the above two inequalities, (3.24) can be deduced. In addition to collecting with Lemma 3.5, we have

This along with (3.3) implies (3.25), then we utilize (3.14) and the above inequality to gain

According to (3.12), we finally infer (3.26).

Lemma 3.7. If $p > 2$, then there exist positive constants $C$ and $C_r = C_r(r, \varepsilon)$ such that

Proof. The inequality (3.27) can be derived by using a Moser-Alikakos-type method presented in ^[21]. Since the process of proof is standard, we omit it. In light of Lemma 2.2, it is straightforward to arrive at (3.28).

Lemma 3.8. If $p > 2$, then we have

where $C > 0$ is a constant.

Proof. By multiplying the first equation of (3.1) by $\frac{\partial n_\varepsilon}{\partial t}$ and integrating by parts over $\Omega$, we can utilize the Young inequality to derive

A combination of Lemmas 3.5 and 3.6 and the above inequality can yield (3.29). Then, for all $t\in(0, T_{max})$, Lemmas 3.6 and 3.7 entail

which paired with the criterion of extensibility in Lemma 3.1 yields $T_{max} = \infty$.

Using the estimates that were gathered in the preceding section, we will show the existence of global weak solutions for system (1.4) in this section.

Proof of Theorem 1.1. If $p > 2$ and $(n_\varepsilon, c_\varepsilon, u_\varepsilon, P_\varepsilon)$ solves (3.1), then we employ the Sobolev compact embedding theorem, Aubin-Lions compactness theorem and Lemmas 3.3, 3.5–3.8, there exists some subsequence of $(\varepsilon_j)_{j\in N}\subset(0, 1)$ with $\varepsilon_j\searrow0$ as $j\rightarrow \infty$ such that

and for any $s\in(1, +\infty)$

To claim that (3.31) holds with $\omega = |\nabla n|^{p-2}\nabla n$, we need to prove that for any $\phi_1\in C^\infty(\bar{\Omega}\times [0, \infty))$,

We replace $\phi_1$ with $n_{\varepsilon}\phi_1$ in (1.6), which implies

Similarly, replacing $\phi_1$ with $n\phi_1$ in (1.6), we can get

For any $\phi_1\ge 0, \zeta\in L^p_{loc}(\mathbb{R}^+; W^{1, p}(\Omega))$, we have

and by moving the item, then we can obtain

By a combination of (3.33) and (3.35), for any $\phi_1\in C^\infty(\bar{\Omega}\times [0, \infty))$ with $\phi_1\ge 0$, we can see that

then we obtain that

Inserting (3.34) into the above inequality, for any $\phi_1\in C^\infty(\bar{\Omega}\times [0, \infty))$ with $\phi_1\ge 0$, we deduce

With $\lambda > 0$, $\psi\in C^\infty(\bar{\Omega}\times [0, \infty)),$ we select $\zeta = n-\lambda\psi$ to arrive at

In the above inequality, letting $\lambda\rightarrow 0$, we attain

Similarly, we take $\zeta = n+\lambda\psi$ to get

So, we derive

Thus, (3.31) is achieved. Letting $\varepsilon\rightarrow 0$ in (1.6)–(1.8), we can deduce

where $\forall\phi_1 \in C^\infty_0(\bar{\Omega}\times [0, \infty))$, $\phi_2 \in C^\infty_0(\bar{\Omega}\times [0, \infty))$, $\psi \in C^\infty_0(\bar{\Omega}\times [0, \infty); \mathbb{R}^3)$ and $\nabla\cdot\psi = 0$. Therefore, the system (1.4) exists a global weak solution $(n, c, u)$. With a combination of (10)–(12), the proof of Theorem 1.1 has been concluded.

4.   Conclusions

In this paper, we considered the chemotaxis-Stokes system (1.4) with slow $p$-Laplacian diffusion and logistic source in a bounded domain $\Omega\subset\mathbb{R}^3$ with zero-flux boundary conditions and no-slip boundary condition and proved the existence of global bounded weak solutions for any slow $p$-Laplacian diffusion $(p > 2)$ under the action of logistic source. The main result is as follows:

Theorem 1.1. If $p > 2$, then the system (1.4) with the initial conditions (1.5) exists global weak solutions $(n, c, u)$ in the bounded domain $\Omega\subset\mathbb{R}^3$, such that for all $\beta\in(\frac{3}{4}, 1), s > 1$, satisfying

where positive constants $M_1, M_2$ only depend on $n_0, c_0, u_0, p, \Omega$. Here $Q(t) = \Omega\times(t, t+1)$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The paper is supported by the Innovation Team Funds of China West Normal University (CXTD2020-5) and the Innovation Project of China West Normal University (cxcy2023054).

Conflict of interest

All authors declare no conflict of interest in this paper.