Global existence of classical solutions for the 2D chemotaxis-fluid system with logistic source

1.   Introduction

The coupled chemotaxis-Navier-Stokes model with logistic source terms is as following: $Q_{T} = (0, T]\times{\Bbb R}^2$

Here the unknows are the concentration of bacteria $n = n(x, t):Q_{T}\rightarrow {\Bbb R}^{+}$; the oxygen concentration $c = c(x, t):Q_{T}\rightarrow {\Bbb R}^{+}$; the given vector field $u = u(x, t):Q_{T}\rightarrow {\Bbb R}^{2}$; and the pressure of the fluid $P = P(x, t):Q_{T}\rightarrow {\Bbb R}$. We denote the corresponding diffusion coefficients for the cells, the oxygen and the fluid by ${\cal D}_{n}, {\cal D}_{c}$ and ${\cal D}_{u}$.

Apart from that, $\chi(c)$ represents the chemotactic sensitivity, and $g(c)$ is the consumption rate of oxygen. In ^[19], it is shown that the functions $g(c)$ and $\chi(c)$ are constants at large $c$ and rapidly approach zero below some critical $c^{\ast}$. Moreover, the experimentalists in ^[19] used multiples of the Heaviside step function to model $\chi(\cdot)$ and $g(\cdot)$. The scalar valued function $\Phi$ is also given.

The model (1.1) was proposed in ^[12], which in order to describe the movement of bacteria in response to the presence of a chemical signal substance (oxygen or another nutrient) in their fluid environment. $\kappa$ is the effective growth rate of the population and $\mu$ controlling death by overcrowding. For more the logistic term related, see ^{[10,11,24,35]}.

In recent years, the study of the dynamics of solutions to (1.1) has attracted many researchers. Finite time blow-up phenomena is one of the most important dynamical problems in (1.1). One can refer to ^[3,30]. It is worth pointing that the logistic term $\kappa n-\mu n^2$ of (1.1) can avoid blow-up phenomena in ^[9].

To make the system (1.1) be well-posed, we add some initial conditions as follows:

Besides, we also have the corresponding chemotaxis-only subsystem of (1.1) on letting $u \equiv 0$, that is, the system

is a variant of the classical Keller-Segel model. Up to now, one notices that the homogeneity of model (1.3) may be undermined by the cross-diffusive term $-\nabla\cdot(n\chi(c)\nabla c)$ and even enforce blow-up of solutions ^[1]. Although we are facing great challenges, there are still numerous analytic results which mainly fixed attention on the local and global solvability of corresponding initial(-boundary)-value problems in either bounded or unbounded domains $\Omega$, under diverse technical conditions on $\chi(\cdot)$ and $g(\cdot)$.

As for initial boundary value problem (1.3) in higher space dimensions, Winkler ^[22] proved the global existence of a small solution to (1.3) and that it asymptotically behaves like the solution of a decoupled system of linear parabolic equations. On the other hand, a result concerning that blow-up behavior occurring for some radially symmetric positive initial data in higher dimension was recently obtained in ^[23]. And one can see a relevant reference in ^[25].

If system (1.3) is coupled with Navier-Stokes equations, we have the following model

As far as we know, results on smooth global solvability could be established for the two-dimensional version of (1.4) whenever $\mu$ is positive (^[18]), while a similar statement could be derived when $\mu\geq23$ in three dimension at least for a Stokes simplification of (1.4) in which the nonlinear convective term $u\cdot\nabla u$ is neglected ^[17]. In a recent paper ^[31], Winkler reveals that whenever $\omega > 0$, requiring that

with some $\eta = \eta(\omega) > 0$, and that $f$ satisfies a suitable assumption on ultimate smallness, is sufficient to ensure that each of these generalized solutions becomes eventually smooth and classical.

For system (1.1), when $\kappa = \mu = 0$, ^[6] demonstrates that there exists a global weak solutions for the 2D chemotaxis-Stokes equations, i.e., the nonlinear convective term $u\cdot\nabla u$ is removed in the fluid equation of (1.1), by making use of quasi-energy functionals associated with (1.1), under the following conditions on $\chi(\cdot)$ and $g(\cdot)$:

which were relaxed in ^[8,13],

for the fully chemotaxis-Navier-Stokes system (1.1) without any smallness conditions on the initial data.

Furthermore, for the smooth initial data, Chae et al. establish the global existence of smooth solutions ^[3] by assuming that $\chi(c)$ and $g(c)$ satisfy

and that there exists a constant $\eta$ such that

which is taken away in ^[4]. For the full 3D chemotaxis-Navier-Stokes system, the global classical solution near constant steady states and the global weak solutions under the special situation that $\chi(\cdot)$ precisely coincides with a fixed multiple of $g(\cdot)$ are constructed in ^[6] and ^[3], respectively.

In the case of $\kappa = \mu = 0$, Winkler establishes the existence of global classical solutions in 2D bounded convex domains $\Omega$ in ^[26]. Then he ^[27] asserts that a solution of the two-dimensional chemotaxis-Navier-Stokes system stabilizes to the spatially uniform equilibrium $(\bar n, 0, 0)$ with $\bar {n_0} = \frac{1}{|\Omega|}\int_\Omega n_0(x)dx$ in the sense of $L^\infty(\Omega)$. If $\kappa\geq0$, $\mu > 0$, in a bounded smooth domain $\Omega\subset{\Bbb R}^3$, J. Lankeit ^[12] shows that after some waiting time weak solutions constructed become smooth and finally converge to the semi-trivial steady state $(\frac{\kappa}{\mu}, 0, 0)$. More related paper on the bounded domain case, for the 2D case, one may refer to ^[16,20,21] and for the 3D case, one also refer to ^{[2,16,21,28,29]}. Additionally, under some strong structural assumptions on $\chi$ and $g$, the global existence of weak solutions is proved as well as their eventual smoothness and stabilization to the 3D version of the chemotaxis-Navier-Stokes system is established in ^[28] and ^[29].

Recently, Duan et al. ^[7] established the global existence of weak solutions and classical solutions for both the Cauchy problem and the initial-boundary value problem supplemented with some initial data. In addition, Wu et al. ^[32] improved the results by taking more careful calculations than those of ^[7].

The main purpose of this paper is to demonstrate the global existence of classical solutions to (1.1) for the Cauchy problem. We first introduce the corresponding blowup criterion.

Now, it is position to state our main theorems in this article.

Theorem 1.1. Let $m\geq3$. Assume that $\chi(\cdot)$, $g(\cdot)\in C^m({\Bbb R})$ with $g(0) = 0$, and that $\|\nabla^{l}\Phi\|_{L^{\infty}({\Bbb R}^2)} < \infty$ for $1 \leq |l|\leq m$.

(A1) Then there exists $T^{\ast} > 0$, the maximal time of existence, such that, if the initial data $(n_{0}, c_{0}, u_{0})\in H^{m-1}({\Bbb R}^2)\times H^m({\Bbb R}^2)\times H^m({\Bbb R}^2;{\Bbb R}^2)$, then there exists a unique classical solution $(n, c, u)$ to system (1.1)-(1.2) satisfying for any $T < T^{\ast}$

and

(A2) Moreover, if the maximal time of existence $T^{\ast} < \infty$, then

The proof of A1 in Theorem 1.1 is standard, one can refer to ^[3,33]. We will focus on proving part A2.

Theorem 1.2. Let $m\geq 3$. Under the assumptions of Theorem 1.1, then system (1.1)-(1.2) admits a unique global-in-time classical solution $(n, c, u)$ satisfying for any $T > 0$

and

Before we are going to prove the main results, we give an important proposition, which can be found in ^[14].

Let

Proposition 1.1. Let the triple $(n_{0}, c_{0}, u_{0})\in X_{0}$, $\nabla\Phi\in L^\infty({\Bbb R}^2)$ and $\chi(\cdot)$, $g(\cdot)\in C^m({\Bbb R})$ with $m\geq3$ and $g(0) = 0$. Then, system (1.1) has a unique global solution $(n, c, u)$ such that

The rest of this article is organized as follows. Let's briefly establish a blow-up criterion in Section 2, discuss the Cauchy problem in Section 3.

2.   Proof of Theorem 1.1

Now, we show the proof of the blow-up criterion for the fluid chemotaxis equations.

Proof. Above all, we take the $L^2$ estimate of $n$ into account. Multiplying $n$ on both sides of $(1.1)_{1}$ and integrating in spaces, by the fact that $\chi$ is continuous and c is uniformly bounded, we deduce

Next, testing $-\Delta c$ to both sides of $(1.1)_{2}$ and making use of the integration by parts, it follows that

Similary, taking the $L^2$ inner product $-\Delta u$ to both sides of $(1.1)_{3}$ and using the integration by parts, we also deduce

From all the estimates, it means that

An application of Gronwall's inequality yields that

Observe that $\|n\|_{L^{\infty}(0, T; L^2)}$ and $\|\nabla n\|_{L^2(0, T; L^2)}$ are uniformly bounded if $\int_{0}^T\|\nabla u\|_{L^{\infty}}+\|\nabla c\|_{L^{\infty}}^2dt$ is bounded. Meanwhile, we notice that $n\in L_{x}^{q}L_{t}^{\infty}$ and $\nabla n^{\frac{q}{2}}\in L_{x}^{2}L_{t}^{2}$ for all $2 < q < \infty$, and, owing to the following deduction

Collecting the above inequality, it ensures that $\|n(t)\|_{L^q}\leq C$, with $C$ is independent of $q$. Then, letting $q\rightarrow \infty$, $n\in L_{x}^{\infty}L_{t}^{\infty}$ is obtained.

Additionally, we consider the estimate in the space $(n, c, u)\in H^{1}\times H^{2}\times H^{2}$. It follows that

In conjunction with Young's inequality and Gronwall's inequality, we conclude

Afterwards, $n\in H_{x}^{1}L_{t}^{\infty}\cap H_{x}^{2}L_{t}^{2}$. For the $H^2$ estimate of c, we get

and

Then we get $c\in H_{x}^2L_{t}^{\infty}\cap H_{x}^3L_{t}^{2}$ and $u\in H_{x}^2L_{t}^{\infty}\cap H_{x}^3L_{t}^{2}$ by Gronwall's inequality. In what follows, we show the estimate in the space $(n, c, u)\in H^2\times H^3\times H^3$. In a similar way, it indicates

We control

and

We also obtained $c\in H_{x}^2L_{t}^{\infty}\cap H_{x}^3L_{t}^{2}$. Consequently, in view of Young's inequality and Gronwall's inequality, it follows that

In what follows, for the estimate of $c$, we further have

For the estimate of $u$, we thereby obtain

Applying Gronwall's inequality, it implies $(c, u)\in(H_{x}^3L_{t}^{\infty}\cap H_{x}^4L_{t}^{2}) \times(H_{x}^3L_{t}^{\infty}\cap H_{x}^4L_{t}^{2})$. Let us demonstrate $H^{m-1}\times H^m\times H^m$ estimates. We've proved that the case $m = 2, 3$ and $4$, and therefore we consider the case $m\geq5$. Operating $\partial^{\alpha}(|\alpha|\leq m-1)$ and multiplying $\partial^{\alpha}n$ on both sides of $(1.1)_{1}$ and integrating in spaces, we thereby obtain

The estimate for the case $m = 4$ was already obtained, thus $\|\nabla c\|_{L^{\infty}(0, T; L^{\infty})}$ is bounded. Subsequently, we derive

the classical product lemma on each step of iteration shows

Therefore we also obtain

applying the product lemma. Add up the inequality above and we get

Similarly, for the $H^m$ estimate of $c$, we get

For the estimate of $u$, we also deduce that

Finally, by summing up all the above estimates and utilizing Gronwall's inequality, the fact $(n, c, u)\in(H_{x}^{m-1}L_{t}^{\infty}\cap H_{x}^{m}L_{t}^{2})\times(H_{x}^{m}L_{t}^{\infty}\cap H_{x}^{m+1}L_{t}^{2})\times (H_{x}^{m}L_{t}^{\infty}\cap H_{x}^{m+1}L_{t}^{2})$ is obtained.

As a matter of fact that $\|\nabla c\|_{L^{\infty}}$ is solely responsible for $n\in L_{x}^2L_{t}^{\infty}$ and $\nabla n\in L_{x}^2L_{t}^2$ by the above process. That is

This yields $u\in L_{x}^2L_{t}^{\infty}$ and $\nabla u\in L_{x}^2L_{t}^{2}$ by

Furthermore, we obtain $n\in L_{x}^qL_{t}^{\infty}$ and $\nabla n^{\frac{q}{2}}\in L_{x}^2L_{t}^{2}$ for all $2 < q < \infty$

Besides, we note that $\nabla c\in L_{x}^2L_{t}^{\infty}$ and $\nabla^2c\in L_{x}^2L_{t}^2$. Actually,

Finally, we denote vorticity as $\omega: = \nabla\times u$; where $\omega = \partial_{1}u_{2}-\partial_{2}u_{1}$ in two dimensions. Then, let us set $\nabla^{\bot}n = (-\partial_{2}n, \partial_{1}n)$. We investigate the vorticity equation

We notice that $\omega\in L_{x}^2 L_{t}^{\infty}$ and $\nabla\omega\in L_{x}^2L_{t}^2$, on account of

In addition, we note that $\nabla\omega\in L_{x}^2L_{t}^{\infty}$ and $\nabla^2\omega \in L_{x}^2L_{t}^2$. In fact, testing $-\Delta\omega$, this implies that

Hence, via embedding, we get

This implies the desired result.

3.   Proof of Theorem 1.2

In this section, we will show that the local classical solutions can be extended at any time $T > 0$.

Proof of Theorem 1.2. The process is similar to the proof of Theorem 1.2 in ^[7]. Using contradictory methods, supposing that the maximal time $T^{\ast}$ is finite, we will prove

which contradicts the extensibility criterion (1.10). Actually, according to the fact

where $\widetilde C_{GN}$ is a positive constant resulted from the Gagliardo-Nirenberg inequality.

Now, we just need to verify that

and

According to Proposition 1, we obtain (3.1).

As for (3.2), applying $\Delta$ to the $(1.1)_{2}$, multiplying $\Delta c$ with the resulted equation, and integrating over ${\Bbb R}^2$, we have

For $I_{1}$, we get

Similarly, let $M = \|c_0\|_{L^\infty}$, we conclude

and

Collecting $I_{1}-I_{3}$, we obtain

with $C_{2}$ is a positive constant depending on the Sobolev's embedding and the initial data. The Gronwall's inequality implies that

for all $t\in(0, T^{\ast})$. Therefore we have verified (3.2) provided that

According to Proposition 1, we verify the above inequality. For convenience of readers, we give a sketch of the proof. To begin with, taking the $L^2$-inner product with $n$ for equation $(1.1)_{1}$, we get

Next, we control $\int_{{\Bbb R}^2}n^2|\nabla c|^2dx$ of the above inequality

with $C_{3} = \frac{\sup_{0\leq s\leq M}\chi^4(s)C_{GN}^2M^2}{{\cal D}_{n}^3}$, by using Sobolev's embedding, Young's inequality and the boundedness of $c$. Therefore, we have

In view of Gronwall's inequality, we deduce

for all $t\in(0, T^{\ast})$ and some positive constant $C_{4}$ depending on the initial data and the maximal time $T^{\ast}$, where we also used the inequality

by recalling Proposition 1.1. A direct integration on $[0, T^{\ast}]$ implies that

Similarly, we also have

Let's first investigate the integrability of the second derivative of $u$. For convenience, let $\omega : = \nabla^{\bot}u$ be the vorticity of $u$ and then the vorticity equation as follow:

Next, a direct energy method follows that

This leads to the vorticity estimate

where $C_{5}$ is a positive constant depending on $\nabla\Phi$. In conjunction with Gronwall's inequality and (3.3), we infer

for all $t\in(0, T^{\ast})$ with $C_{6}: = \int_{{\Bbb R}^2}|\omega|^2(x, 0)dx+C_{4}C_{5}T^{\ast}$. Application of the above inequality and the Biot-Savart law, we derive

Sum up (3.3)-(3.5), it implies the desired result. This completes the Proof of Theorem 1.2.

Acknowledgments

Q. Zhang was partially supported by the Natural Science Foundation of Hebei Province [grant number A2020201014 and A2019201106]; the Second Batch of Young Talents of Hebei Province; Nonlinear Analysis Innovation Team of Hebei University.

Conflict of interest

All authors declare that there is no interests in this paper.

A.   Appendix

In this appendix, we will give a sketch proof of local existence.

We first introduce the dynamic partition of the unity to define Besov spaces. One may check ^[15] for more details. Let $\varphi\in C_{0}^{\infty}(\mathbb{R}^{d})$ be supported in ${\cal C} = \{\xi\in\mathbb{R}^d, \, \frac{3}{4}\le|\xi|\le\frac{8}{3}\}$ such that

Defining $\chi(\xi) = 1-\sum\limits_{q\in N}\varphi(2^{-q}\xi)$. For $f\in{\cal S'}$, we set Littlewood-Paley operators as follows

The following low-frequency cut-off will be also used:

Now, let us recall the definition of the Besov space. For $s\in \mathbb{R}$, $1\le p, r\le\infty$, we define the homogenous Besov space $\dot{B}_{p, r}^{s}$ as the set of tempered distributions of $f\in{\cal S'}/{\cal P}$ such that

where ${\cal P}$ is the polynomial space. The inhomogeneous space $B_{p, r}^{s}$ is the set of tempered distribution $f$ such that

It is worthwhile to remark that $B_{2, 2}^{s}$ and $B_{\infty, \infty}^{s}$ coincide with the usual Sobolev spaces $H^{s}$ and the usual Hölder space $C^{s}$ for $s\in \mathbb{R}\setminus \mathbb{Z}$, respectively.

In our study, we require the space-time Besov spaces as following manner: for $T > 0$ and $n\ge1$, we denote by $L_{T}^{\rho}{B}_{p, r}^{s}$ the set of all tempered distribution $f$ satisfying

Lemma A.1. ^[15] Let $1\le p\le q\le\infty$. Assume that $f\in L^p$, then there exists a constant $C$ independent of $f$, $j$ such that

Lemma A.2. ^[15] There exists a constant $C > 0$ such that for $S > 0$, we have

Lemma A.3. ^[5] Let $u$ be a solution of the transport equation

and define $R_{q}: = v\cdot\nabla\Delta_{q}u-\Delta_{q}(v\cdot\nabla u)$, $1\le p\le p_{1}\le\infty$, $1\le r\le\infty$ and $s\in \mathbb{R}$ such that

There exists a sequence $c_{q}\in \ell^{r}(\mathbb{Z})$ such that $\|c_{q}\|_{\ell^{r}} = 1$ and a constant $C$ depending only on $d, r, s, p$, and $p_{1}$, which satisfy

with

Theorem A.4. Let $s\geq2$. Assume that $\chi(\cdot)$, $g(\cdot)\in C^s(\mathbb{R})$ with $g(0) = 0$, and that $\|\nabla^{l}\Phi\|_{L^{\infty}(\mathbb{R}^2)} < \infty$ for $1 \leq |l|\leq s$.

(A1) Then there exists $T^{\ast} > 0$, the maximal time of existence, such that, if the initial data $(n_{0}, c_{0}, u_{0})\in H^{s-1}(\mathbb{R}^2)\times H^s(\mathbb{R}^2)\times H^s(\mathbb{R}^2;\mathbb{R}^2)$, then there exists a unique classical solution $(n, c, u)$ to system (1.1)-(1.2) satisfying for any $T < T^{\ast}$

and

Proof. We construct the following regularized system:

Step 1. Uniform estimates.

Taking the operation $\Delta_{q}$ with $q\ge-1$ on the first equation of (A.1), we have

Multiplying (A.2) by $\Delta_{q}n^{k}$ and integrating by parts gives

Multiplying $2^{2qs}$ on both sides of the above inequality, then taking the $\ell^{1}$ norm, using Hölder's inequality and Young's inequality together with Lemma 4.2, we get

from which we have

In a similarly way to (A.3), we obtain

it follows that

Applying $\Delta_{q}$ with $q\ge-1$ to the third equation of (A.1) yields

Multiplying the above equality with $\Delta_{q}u^{k}$ yields

Multiplying $2^{2q(s+1)}$ on both side of the above inequality and taking the $l^{1}$ norm, we obtain

Summing up (A.3)-(A.5), we obtain

We conclude from the Gronwall inequality that

We can choose

such that

Step 2. Compactness.

From (A.6), we obtain

In order to show the convergence, we also need uniform boundedness for $\partial_{t}n^{k}$, $\partial_{t}c^{k}$ and $\partial_{t}u^{k}$. From the first equation of (A.1), we know

Similarly, we have

and

Since $L^{2}$ is locally compactly embedded in $H^{-1}$, we apply the Aubin-Lions Lemma to conclude that, up to an extraction of subsequence, the approximate solution sequence $(n^{k}, c^{k}, u^{k})$ strongly converges in $L^{\infty}([0, T]; H^{-1})$ to some function $(n, c, u)$ such that

Using the above estimates, it is easy to pass the limit in the approximate system (A.1) and $(n, c, u)$ solve (1.1) in the sense of distribution. By a classical deduction ^[34], we get $n\in {\cal C}([0, T]; H^{s})$, $c\in {\cal C}([0, T]; H^{s+1})$ and $u\in {\cal C}([0, T]; H^{s+1})$.

By virtue of Heat equation theory, we can prove the time differentiation. For example, we consider the following equations:

Suppose $u\in L_t^\infty H^s$ and $f\in L^\infty_t H^{s-2}$, we obtain $u_t\in L^\infty_t H^{s-2}$. For the arbitrariness of $s$, we have $u_t\in L^\infty_tH^s, \forall\ s > 0$. In a similar way, we get

and $u_{tt}\in L^\infty_t H^{s-2}$. Thus we show the time differentiation.

Step 3. Uniqueness.

Let us consider the two solutions $(n_{1}, c_{1}, u_{1})$ and $(n_{2}, c_{2}, u_{2})$ associated with the same initial data and satisfy (1.1). We use the notation $\delta n = n_{1}-n_{2}$, $\delta c = c_{1}-c_{2}$ and $\delta u = u_{1}-u_{2}$. Then we have

Taking the $L^{2}$-inner product of the first equation with $\delta n$, we have

from which we get

Next, taking the $L^{2}$-inner product of the second equation of Eqs.(A.7) with $\delta c$, we know

Hence we get

Taking $\partial_{i}$ on both sides of the second equation of Eqs (A.7) yields

Multiplying the above equation with $\partial_{i}\delta c$ and integrating with respect to space variable, we obtain

Then we get

Performing the $L^{2}$-inner product of the third equation of system (A.7) with $\delta u$, we get

Such that

From ((A.8)-(A.11), we have

Consequently, we have

where

By the above estimates, we know that $F(t)$ is integrable. Therefore, we finally obtain the uniqueness by using the Gronwall inequality.