Trust and commitment in supply chain during digital transformation: A case in Korea

1.   Introduction and main results

Chemotaxis is a biological phenomenon which describes the oriented movement of cells (or organisms) in response to chemical gradients ^[1,2]. As early as 1970, Keller and Segel ^[3] originally introduced a chemotaxis model through a system of parabolic equations. This model reads

where $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary, the unknown functions $n$ and $c$ respectively represent the cell density and the signal concentration, and $S$ denotes the chemotactic sensitivity. This model is primarily used to describe the aggregation phenomenon of Dictyostelium discoideum, where the effects of the chemical signal secreted by themselves are taken into consideration. During the past half a century, the Keller-Segel model has been attracting many scholars' attention. The known results are concentrated on whether the solutions for Neumann boundary problem of (1.1) globally exist or blow up in finite time. Concretely, if $S: = S(n)$ is a scalar function fulfilling $S(n)\leq C_{S}(n+1)^{-\alpha}$ with some $C_{S} > 0$ and $\alpha > 0$, then for all $\alpha > 1-\frac{2}{N}$, the corresponding problem has a global solution which is uniformly bounded ^[4]. However, if $S$ satisfies $S(n) > c_{S}n^{-\alpha}$ with some $c_{S} > 0$ and $\alpha < 1-\frac{2}{N}$ for $N\geq2$, and $\Omega$ is a ball, then the solution of (1.1) will blow up in finite time. So,

is called the critical exponent of the blow-up phenomenon. Recently, some results relating to the well-posedness of the hyperbolic Keller-Segel equation in the Besov framework were obtained in ^[5]. Afterwards, Zhang et al. ^[6] improved these results and established two kinds of blow-up criteria of strong solutions in Besov spaces by means of Littlewood-Paley theory. For more results about (1.1) and its variations, we refer interested readers to ^{[7,8,9,10,11,12,13,14,15,16]}

If we consider the framework where the chemical is produced by the cells indirectly, the corresponding chemotaxis model turns to the following Keller-Segel system with indirect signal production:

in a bounded domain $\Omega\subset \mathbb{R}^{N}$ with smooth boundary, where the functions $n$, $v$ and $w$ represent the cells, density, the signal and the chemical concentration, respectively. If $S(x, n, v, w) = \chi$ with $\chi > 0$ and $N\leq3$, Fujie and Senba ^[17] showed that the homogeneous Neumann (or mixed) boundary problem of system (1.2) possesses a unique and globally bounded classical solution.

However, in many cases, the migration of cells (or bacteria) is largely affected by their surrounding environment ^[18,19]. If the cells consume the chemical signal, Tuval et al. ^[19] introduced the following chemotaxis-fluid system:

in a bounded domain $\Omega \subset \mathbb{R}^3$ with smooth boundary, where $f(c)$ measures the rate that cells consume the oxygen, and $S(x, n, c)$ is a tensor-valued (or scalar) chemotactic sensitivity. Most remarkably, by means of the chemical consumption setting and the maximum principle of the parabolic equations, we may directly deduce that $c$ is uniformly bounded from the second equation of (1.3). This significant feature leads to the chemotaxis-fluid model with the framework of signal consumption being more intensively studied than the signal production mechanism. For instance, Winkler ^[20] proved that the global weak solution of system (1.3) which has enough regularity properties and thereby fulfills the condition of so-called eventual energy solution (this concept is newly proposed in his paper) becomes eventually smooth after some waiting time. For more studies about this system, one can refer to Zheng ^[21], Winkler ^{[22,23,24,25]} and other results on the global solvability and asymptotic behavior, such as ^[26,27,28], for details.

Considering the framework where the chemical signal is produced by the cells instead of consuming it, the corresponding chemotaxis-fluid model becomes the following Keller-Segel(-Navier)-Stokes system:

where $S$ is a tensor-valued (or scalar) function, and $\Omega \subset \mathbb{R}^{N}$ with smooth boundary. Let us just list a few representative results. For the Navier-Stokes fluid (i.e. $\kappa \neq 0$), if $|S(x, n, c)|\leq C_{S}(n+1)^{-\alpha}$ with $C_{S} > 0$ and $\alpha > 0$, in 2D case Wang et al. ^[29] showed that the initial-boundary value problem of (1.4) admits at least one classical solution. In the 3D Stokes case (i.e. $\kappa = 0$) of (1.4), Wang and Xiang ^[30] got the same results for $\alpha > \frac{1}{2}$. For the 3D Navier-Stokes version of system (1.4), Liu and Wang ^[31] verified that there exists at least one global weak solution for the corresponding initial-boundary value problem of (1.4) if $|S(x, n, c)|\leq C_{S}(n+1)^{-\alpha}$ with some $C_{S} > 0$ and $\alpha > \frac{3}{7}$. Recently, Ke and Zheng ^[32] improved the restriction admitting a global weak solution from $\alpha > \frac{3}{7}$ to $\alpha > \frac{1}{3}$, which compared with the known result of the fluid-free system is an optimal restriction on $\alpha$. As for the further results, under assumption $\alpha\geq1$ and an explicit condition on the size of $C_{S}$, Zheng ^[33] confirmed that the weak solution of system (1.4) would be eventually smooth, and that it is close to a unique spatially homogeneous steady state. Additionally, one can see ^[34,35,36] and the references therein to find more conclusions about this system.

Motivated by the above works, in this paper we consider the following initial-boundary value problem of the Keller-Segel-Navier-Stokes system with indirect signal production:

where $\Omega \subset \mathbb{R}^{2}$ is a bounded domain with smooth boundary, and $S(x, n, v, w)$ satisfies

and

with some $C_{S} > 0$ and $\alpha\geq0$. To state our main results of this paper, we make the following assumptions that

and the initial data $\left(n_{0}, v_{0}, w_{0}, u_{0}\right)$ satisfies

where $A_r$ denotes the Stokes operator with domain $D(A_r): = W^{2, r}(\Omega)\cap W_0^{1, r}(\Omega)\cap L_\sigma^r(\Omega)$ and $L_\sigma^{r}(\Omega): = \{\varphi \in L^r(\Omega)|\nabla\cdot\varphi = 0\}$ ^[37].

With these assumptions at hand, we can state the following main results.

Theorem 1.1. If (1.6), (1.7), (1.8) and (1.9) hold, then for any

there exists a global classical solution $(n, v, w, u, P)$ of problem (1.5) which fulfills

with $p, q > 1$ and $\gamma\in(0, 1)$, where $n$, $v$ and $w$ are nonnegative in $\Omega\times(0, \infty)$. Moreover, the solution is bounded, and there exists $C(\gamma) > 0$ such that

for all $t > 0$, where $\gamma$ is given by (1.9).

Remark 1.1. Theorem 1.1 improves the result of Yu ^[38], which showed that if $\alpha > 0$, then the Stokes version of problem (1.5) possesses a global classical solution.

This paper is organized as follows. In Section 2, we claim that the regularized problem possesses at least one local classical solution which is nonnegative. Relying on a series of $\varepsilon$-independent a priori estimates obtained in Section 3, in Section 4 we verify the local existence of a classical solution for regularized problem can be extended to the global. In Section 5, we will construct a global weak solution which has enough regularity to become a classical solution to prove our main results.

2.   Preliminaries

Compared with the classical Keller-Segel model, the convection term presenting in the Navier-Stokes equations engenders more mathematical difficulties. We define

where $\{\rho_{\varepsilon}\}_{\varepsilon \in (0, 1)} \in C_0^{\infty}(\Omega)$ is a family of standard cut-off functions fulfilling $0 \leq \rho_{\varepsilon} \leq 1 \text{ in } \Omega$ and $\rho_{\varepsilon} \nearrow 1$ in $\Omega$ as $\varepsilon \searrow 0$, and $\chi_{\varepsilon} \in C_0^{\infty}([0, \infty))$ satisfies $0 \leq \chi_{\varepsilon} \leq 1$ in $[0, \infty)$ and $\chi_{\varepsilon} \nearrow 1$ in $[0, \infty)$ as $\varepsilon \searrow 0$.

Then, we can introduce the following approximate system of (1.5):

where

is the standard Yosida approximation, and $A: = -\mathcal{P}\Delta$ is the realization of the Stokes operator with $\mathcal{P}$ denoting the Helmholtz projection of $L^2(\Omega)$ onto solenoidal subspace $L^2_\sigma(\Omega)$ ^[37].

Our main idea is to construct a weak solution which is globally bounded (the concept of weak solution can be found in Definition 5.1), and we claim it possesses adequate regularity to be a classical solution. The biggest obstacle we must deal with is the bad regularity of $n$ caused by the small exponent $\alpha$. Our main tool is based upon an energy estimate concerning the functional

which successfully overcomes this difficulty. The appropriately regularized problem (2.2) possesses local-in-time classical solution, which can be stated as follows.

Lemma 2.1. Suppose $\phi \in W^{2, \infty}(\Omega)$ and $\varepsilon \in(0, 1)$. Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with smooth boundary. Then, there exist $T_{max, \varepsilon} \in(0, \infty]$ and a classical solution $\left(n_{\varepsilon}, v_{\varepsilon}, w_{\varepsilon}, u_{\varepsilon}, P_{\varepsilon}\right)$ of (2.2) in $\Omega \times\left(0, T_{max, \varepsilon}\right)$ such that

solves (2.2) in the classical sence in $\Omega\times\left[0, T_{max, \varepsilon}\right)$, and that $n_{\varepsilon}$, $v_{\varepsilon}$ and $w_{\varepsilon}$ are nonnegative in $\Omega \times\left(0, T_{max, \varepsilon}\right)$. Moreover, if $T_{max, \varepsilon} < \infty$, then we have

as $t\rightarrow T_{max, \varepsilon}$, where $\gamma$ is similar to that in (1.9).

Proof. The fixed point argument which is established in ^[39,40] enables us to immediately substantiate the local existence for a classical solution which complies with (2.3). That $n_{\varepsilon}$, $v_{\varepsilon}$ and $w_{\varepsilon}$ are nonnegative is a clear conclusion of the maximum principle ^[41].

3.   A priori estimates

In this section, we will derive a series $\varepsilon$-independent a priori estimates of the classical solution $(n_{\varepsilon}, v_{\varepsilon}, w_{\varepsilon}, u_{\varepsilon})$ of regularized problem (2.2) from Lemma 2.1. By the way, we take $\tau = \mathrm{min}\left\{1, \frac{1}{4} T_{max, \varepsilon}\right\}$. The positive constants $C_i\ (i\in\mathbb{N}^{\ast})$ appearing in the proof of every lemma are independent of $\varepsilon\in(0, 1)$, which only depend on $\Omega, \alpha, C_{S}, \phi, n_0, v_0, w_0$ and $u_0$ if there is no especial explanation. Firstly, by simple integration and ODE comparison arguments, we obtain the following boundedness of $L^1$-norms, which is common in many chemotaxis models.

Lemma 3.1. For any $\varepsilon \in(0, 1)$, the solution of (2.2) satisfies

as well as

and

Lemma 3.2. If $\alpha > 0$, then for any $\mu > 0$, there exists a constant $C > 0$ independent of $\varepsilon \in (0, 1)$ such that

Proof. For any $\mu > 0$, the Gagliardo-Nirenberg inequality and the Young inequality provide positive constants $C_1$ and $C_2$ such that

where the boundedness of $\left\| n_{\varepsilon} \right\|_{L^1(\Omega)}$ from (3.1) and the fact that $\frac{2}{1+\alpha} < 2$ by $\alpha > 0$ are used.

Lemma 3.3. If $\alpha > 0$ and $p\in [2, 2+\frac{2\alpha}{1+\alpha})$, then there exists some $C > 0$ such that for all $\varepsilon \in (0, 1)$ the solution of (2.2) satisfies

and

Proof. This proof may be divided into two cases:

Case 1, $\alpha\neq\frac{1}{2}$.

First, multiplying the first equation of (2.2) by $n_{\varepsilon}^{2\alpha-1}$, employing the fact that $\nabla \cdot u_ {\varepsilon} = 0$, integrating by parts, we derive

Applying the Young inequality and the trivial fact $\frac{n_{\varepsilon }}{n_{\varepsilon}+1}\leq1$, by (1.7), we obtain

Namely,

Next, testing the second equation of (2.2) by $v_{\varepsilon}$, utilizing the fact that $u_ {\varepsilon}$ is divergence-free and the continuous embedding $W^{1, 2}(\Omega)\hookrightarrow L^{\frac{1+\alpha}{\alpha }}(\Omega)$, by virtue of the Hölder inequality and the Cauchy-Schwarz inequality, we deduce that there exists a constant $C_1 > 0$ satisfying

Hence,

This along with a multiple of (3.10) yields that

Then, multiplying the third equation of (2.2) by $w^{p-1}_{\varepsilon}$ with $p\in [2, 2+\frac{2\alpha}{1+\alpha})$, integrating by parts, applying the Hölder and the Cauchy-Schwarz inequalities as well as the fact $\nabla \cdot u_ {\varepsilon} = 0$, we have

Combining (3.13) and (3.14), we get

To handle these three terms on the right side of (3.15), one can employ the Gagliardo-Nirenberg and the Young inequalities as well as Lemma 3.2 to estimate

as well as

and

with positive constants $C_2$, $C_3$, $C_4$, $C_5$ and $C_6$, where we have $\frac{4\alpha}{(1+\alpha)p} < 2$ and $\frac{4}{p}\left(p-1-\frac{\alpha}{1+\alpha}\right) < 2$ by $p \in[2, 2+\frac{2\alpha}{1+\alpha})$. Substituting (3.16), (3.17) and (3.18) into (3.15), one has

with $C_7: = C_3+C_5+C_6$. If $sgn(2\alpha-1) = 1$ (i.e. $\alpha > \frac{1}{2}$), (3.19) in conjunction with some standard arguments implies that (3.6) and (3.7) hold. On the other hand, if $sgn(2\alpha-1) = -1$ (i.e. $0 < \alpha < \frac{1}{2}$), we set

and

By the Gagliardo-Nirenberg estimate

and (3.19), we deduce

with $C_{10}: = C_7+C_{9}$, where we observe the fact that $f_{\varepsilon}(t)\leq \int_{\Omega} v_{\varepsilon}^{2}(\cdot, t)+\frac{1}{p}\int_{\Omega}w_{\varepsilon}^{p}(\cdot, t)$. In view of an ODE comparison argument, from (3.22), we obtain a constant $C_{11} > 0$ such that

Since $0 < \alpha < \frac{1}{2}$, the boundedness of $\int_{\Omega}n_{\varepsilon}^{2\alpha}$ is an immediate consequence by (3.1). Thus, (3.23) guarantees the existence of some constant $C_{12} > 0$ satisfying

Integrating (3.19) in time, there exists a $C_{13} > 0$ fulfilling

Consequently, (3.6) and (3.7) hold for $\alpha\neq\frac{1}{2}$.

Case 2, $\alpha = \frac{1}{2}$.

By the first equation of (2.2), one may exploit the Young inequality to estimate

That is,

Using the same arguments as proving case $0 < \alpha < \frac{1}{2}$, it is deduced that

and

with positive constants $C_{14}$ and $C_{15}$. Thus, (3.6) and (3.7) hold for $\alpha = \frac{1}{2}$.

Therefore, we may merge these two cases to conclude that (3.6) and (3.7) hold for any $\alpha > 0$. We complete this proof.

Lemma 3.4. There exists a constant $C > 0$ independent of $\varepsilon \in (0, 1)$ such that the solution of (2.2) satisfies

and

Proof. Multiplying the fourth equation of (2.2) by $u_{\varepsilon}$, recalling the fact that $u_ {\varepsilon}$ is divergence-free, integrating by parts, we have

Then, in light of the Hölder inequality as well as the Young inequality and the continuous embedding $W^{1, 2}(\Omega) \hookrightarrow L^\frac{1+\alpha}{\alpha}(\Omega)$, we apply Lemma 3.2 to estimate

with positive constants $C_{1}$, $C_{2}$ and $C_{3}$. Now, inserting (3.33) into (3.32) and considering the estimate obtained by (3.7), we obtain

and

with positive constants $C_{4}$ and $C_{5}$. The proof is completed.

By almost exactly analogous argument with Lemma 6.1 in ^[29], one can directly derive the higher norm estimate of $w_{\varepsilon}$.

Lemma 3.5. For any $q\geq2$, one can find a constant $C > 0$ independent of $\varepsilon \in (0, 1)$ such that

Lemma 3.6. For any $\varepsilon \in (0, 1)$, there exists a constant $C > 0$ that satisfies

and

Proof. Testing the second equation in (2.2) by $-\Delta v_{\varepsilon}$, by applying the Young inequality and integrating by parts, we derive

where the positive constant $C_1$ satisfies $\int_{\Omega}w_{\varepsilon}^2\leq C_1$, and we have the fact that

In view of the standard elliptic regularity theory, the Gagliardo-Nirenberg inequality and the Young inequality provide a constant $C_2 > 0$ fulfilling

This in conjunction with (3.39) indicates that

If we put

and

then (3.42) yields that

where

Recalling the estimates inferred from (3.7) and (3.31), there are two positive constants $C_3$ and $C_4$ satisfying

and

Furthermore, for any $t\in \left(0, T_{max, \varepsilon}\right)$, one can pick a $t_0 \in[(t-\tau)_{+}, t)$ such that $y_{\varepsilon}(\cdot, t_0)\leq C_5$ with some $C_5 > 0$. Invoking the Gronwall inequality, we obtain

which implies (3.37). Finally, integrating (3.42) in time and exploiting the estimates obtained in (3.37) and (3.31), we can verify (3.38) is valid.

Lemma 3.7. If $\alpha > 0$, then there exists a constant $C > 0$ independent of $\varepsilon \in (0, 1)$ such that

and

Particularly, one has

Proof. Testing the first equation of (2.2) by $n_{\varepsilon}^{\alpha}$, noticing the fact that $\nabla \cdot u_{\varepsilon} = 0$ and integrating by parts, by using the Young inequality and (1.7), we arrive at

where one can readily see that $n_{\varepsilon}^{2\alpha}(n_{\varepsilon}+1)^{-4\alpha} = \left(\frac{n_{\varepsilon }}{n_{\varepsilon}+1}\right)^{2\alpha}\left(n_{\varepsilon}+1\right)^{-2\alpha} < 1$ by $\alpha > 0$. By means of the Gagliardo-Nirenberg inequality and the Young inequality, we conclude that

with positive constants $C_1$ and $C_2$, where we observe the truth that $\frac{2}{1+\alpha} < 2$ by $\alpha > 0$. Moreover, looking back on the estimate in (3.37), we utilize the Gagliardo-Nirenberg inequality and the elliptic regularity to ensure the existence of constants $C_3 > 0$ and $C_4 > 0$ such that

Accordingly, (3.48) in combination with (3.49) and (3.50) leads to

Recalling the spatio-temporal boundedness of $\int_{t}^{t+\tau} \int_{\Omega}\left|\Delta v_{\varepsilon}\right|^{2}$ inferred from (3.38), (3.51) implies (3.45) and (3.46). Finally, integrating (3.49) in time, (3.46) yields (3.47).

Relying on the spatio-temporal estimates of $\int_{t}^{t+\tau} \int_{\Omega}\left|\nabla u_{\varepsilon}\right|^{2}$ (see Lemma 3.4) and $\int_{t}^{t+\tau}\int_{\Omega} n^2_{\varepsilon}$ (see Lemma 3.7), one can improve the regularity features of the corresponding fluid field. Since the proof may be found in many papers ^[42], the details are omitted in order to avoid duplication.

Lemma 3.8. There exists some $C > 0$ such that for all $\varepsilon \in (0, 1)$ the solution of (2.2) satisfies

4.   The global solvability of regularized problem (2.2)

In this section, we will prove the local-in-time solutions of regularized problem (2.2) are actually global. Without loss of generality, in this section we presume $0 < \alpha < \frac{1}{2}$. If $\alpha\geq\frac{1}{2}$, at least the boundedness of $\|n_{\varepsilon}\|_{L^{\frac{3}{2 }}(\Omega)}$ can be deduced from (3.45). With the higher regularity of $n_{\varepsilon}$, it becomes easier than case $0 < \alpha < \frac{1}{2}$ to get our desired conclusion. Thanks to the well-known smoothing properties of the Stokes semigroup and the Neumann heat semigroup, one can derive the following uniform $L^{\infty}$ estimates for $n_{\varepsilon}$, $\nabla v_{\varepsilon}$, $\nabla w_{\varepsilon}$ and $u_{\varepsilon}$.

Lemma 4.1. If $\alpha > 0$ and $\gamma \in\left(\frac{1}{2}, 1\right)$, then there exists a constant $C > 0$ such that for all $\varepsilon \in (0, 1)$, the classical solution of (2.2) satisfies

Moreover, for $p > 1$, we can find a positive constant $C(p)$ such that

Proof. For the sake of clarity, this proof is divided into several steps. It is worth mentioning that the following constants $C_i\ (i\in\mathbb{N}^{\ast})$ are independent of $\varepsilon \in (0, 1)$.

Step 1. The boundedness of $\|\nabla w_{\varepsilon}(\cdot, t)\|_{L^{\frac{2}{1-\alpha }}(\Omega)}$ and $\|\nabla v_{\varepsilon}(\cdot, t)\|_{L^{\tilde {p }}(\Omega)}$ with $\tilde {p} > 2$ for all $t \in\left(0, T_{max, \varepsilon}\right)$.

Since $0 < \alpha < \frac{1}{2}$, we have $\frac{2}{1-\alpha} > 2$. First, utilizing the variation-of-constants formula for $w_{\varepsilon}$, we obtain

With the boundedness of $\|n_{\varepsilon}(\cdot, s)\|_{L^{1+\alpha}(\Omega)}$ obtained by Lemma 3.7, in view of the $L^p-L^q$ estimates associated heat semigroup, we deduce

and

with $\lambda > 0$, where we have the fact that $-\frac{1}{2}-\left(\frac{1}{1+\alpha}-\frac{1-\alpha}{2}\right) > -1$ by $0 < \alpha < \frac{1}{2}$. Furthermore, taking $\varsigma = \frac{19}{40}$ and $\delta = \frac{1}{80}$ so that $\frac{1}{2}+\left(\frac{1}{5}-\frac{1-\alpha}{2}\right) < \varsigma$ and $-\varsigma-\frac{1}{2}-\delta > -1$, we can infer that

where the boundedness of $\left\|u_{\varepsilon}(\cdot, s)\right\|_{L^{10}(\Omega)}$ is derived from Lemma 3.8 along with the continuous embedding $W^{1, 2}(\Omega)\hookrightarrow L^{10}(\Omega)$, and $\left\|w_{\varepsilon}(\cdot, s)\right\|_{L^{10}(\Omega)}$ is ensured by Lemma 3.5. Therefore, by accumulating (4.3)-(4.6), the boundedness of $\|\nabla w_{\varepsilon}(\cdot, t)\|_{L^{\frac{2}{1-\alpha }}(\Omega)}$ is obtained. With some very similar arguments, one can derive the boundedness of $\|\nabla v_{\varepsilon}(\cdot, t)\|_{L^{\tilde {p }}(\Omega)}$ with some $\tilde {p} > 2$.

Step 2. The boundedness of $\|n_{\varepsilon}(\cdot, t)\|_{L^{\infty}(\Omega)}$ for all $t \in\left(0, T_{max, \varepsilon}\right)$.

Letting

and

then by the $L^{\tilde {p }}$ estimate of $\nabla v_{\varepsilon}$, we obtain

Exploiting the associate variation-of-constants formula for $n_{\varepsilon}$, in light of the fact that $\nabla \cdot u_{\varepsilon} = 0$, we obtain

If $0 < t\leq1$, then in view of the maximum principle, we have

If $t > 1$, then by the $L^p-L^q$ estimates of the Neumann heat semigroup, we deduce

Next, fixing $q \in\left(2, \tilde {p}\right)$, we may utilize the well-known smoothing properties of the Neumann heat semigroup and the Hölder inequality to conclude

where $\sigma: = \frac{q \tilde {p}-\tilde {p}+q}{q \tilde {p }} \in(0, 1)$, and $-\frac{1}{2}-\frac{2}{2 q} > -1$ by $q > 2$. Collecting (4.7)–(4.11) and utilizing the definition of $M(T)$, there is a $C_{12} > 0$ such that

Since $\sigma < 1$, by some basic calculation we have

Step 3. The boundedness of $\|u_{\varepsilon}(\cdot, t)\|_{L^{\infty}(\Omega)}$ and $\|A^{\gamma}u_{\varepsilon}(\cdot, t)\|_{L^2(\Omega)}$ for all $t \in\left(0, T_{max, \varepsilon}\right)$.

Employing the Helmholtz projection $\mathcal{P}$ to the fourth equation in (2.2), we get the variation-of-constants formula of $u_{\varepsilon}$

where $h_{\varepsilon}(\cdot, s) = \mathcal{P}\left[n_{\varepsilon}(\cdot, s) \nabla \phi-\kappa\left(Y_{\varepsilon} u_{\varepsilon}(\cdot, s) \cdot \nabla\right) u_{\varepsilon}(\cdot, s)\right]$. With the standard smoothing properties of the Stokes semigroup, we derive that for all $t \in\left(0, T_{max, \varepsilon}\right)$ and any $\gamma \in\left(\frac{1}{2}, 1\right)$, there exist $C_{14} > 0$ and $C_{15} > 0$ fulfilling

Choosing $p_0 \in \left(\frac{2}{3-2\gamma}, 2\right)$ such that

the $L^\infty$-estimate of $n_{\varepsilon}$ provides a $C_{16} > 0$ fulfilling

Next, considering the embedding $W^{1, 2}(\Omega) \hookrightarrow L^{\frac{2 p_0}{2-p_0 }}(\Omega)$ and the boundedness of $\left\|\nabla u_{\varepsilon}(\cdot, t)\right\|_{L^2(\Omega)}$ (see Lemma 3.8), we employ the Hölder inequality and the fact that $\mathcal{P}$ is continuous in $L^p\left(\Omega; \mathbb{R}^2\right)$ to achieve that

where we notice the fact that

Assembling (4.12), (4.13) and (4.14), we conclude that

which in combination with the continuous embedding $D\left(A^\gamma\right)\hookrightarrow L^{\infty}(\Omega)$ by $\gamma \in \left(\frac{1}{2}, 1\right)$ yields that

Step 4. The boundedness of $\|\nabla u_{\varepsilon}(\cdot, t)\|_{L^{p}(\Omega)}$ with $p > 1$ for all $t \in\left(0, T_{max, \varepsilon}\right)$.

For any $p > 1$, we can pick suitable $\gamma\in\left(\frac{1}{2}, 1\right)$ satisfying $\gamma > 1-\frac{1}{p}$. By means of the embedding $D(A^{\gamma})\hookrightarrow W^{1, p}(\Omega; \mathbb{R}^2)$ (see ^[37]), (4.2) holds.

Step 5. The boundedness of $\|w_{\varepsilon}(\cdot, t)\|_{W^{1, \infty}(\Omega)}$ and $\|v_{\varepsilon}(\cdot, t)\|_{W^{1, \infty}(\Omega)}$ for all $t \in\left(0, T_{max, \varepsilon}\right)$.

Fixing $\theta \in\left(\frac{1}{2}+\frac{1-\alpha}{2}, 1\right)$, the domain of the fractional power $D\left((-\Delta+1)^\theta\right)$ can be embedded into $W^{1, \infty}(\Omega)$ ^[4]. Accordingly, exploiting the $L^p-L^q$ estimates associated heat semigroup, one has

with $\tau_0 \in\left(0, T_{max, \varepsilon }\right)$. An application of the local solvability of (2.2) indicates that for some $C_{28} > 0$,

Meanwhile, a similar argument yields a $C_{29} > 0$ satisfying

The proof is completed. With the uniform $L^{\infty}$ bounds of $n_{\varepsilon}$, $\nabla v_{\varepsilon}$, $\nabla w_{\varepsilon}$ and $u_{\varepsilon}$ at hand, we claim that the local classical solution of regularized problem (2.2) which is constructed in Lemma 2.1 can be extended to the global.

Proposition 4.1. Let $\alpha > 0$, $\gamma \in\left(\frac{1}{2}, 1\right)$. Let $\left(n_{\varepsilon}, v_{\varepsilon}, w_{\varepsilon}, u_{\varepsilon}, P_{\varepsilon}\right)_{\varepsilon \in(0, 1)}$ be classical solutions of (2.2) constructed in Lemma $2.1$ on $\left[0, T_{max, \varepsilon}\right)$. Then, we have $T_{max, \varepsilon} = \infty$. Moreover, one can find a $C > 0$ which is independent of $\varepsilon \in(0, 1)$ such that

In addition, there is a $C(p) > 0$ fulfilling

As the straightforward result of Proposition 4.1, in light of the standard parabolic regularity (see e.g. Lemmata 3.18 and 3.19 in ^[43]), we can get the following Hölder continuity of $v_{\varepsilon}, \nabla v_{\varepsilon}$ as well as $w_{\varepsilon}, \nabla w_{\varepsilon}$ and $u_{\varepsilon}$.

Lemma 4.2. If $\alpha > 0$, then there exist $\mu \in(0, 1)$ and some $C > 0$ such that

Moreover, for any $\tau > 0$, one can find a $C(\tau) > 0$ satisfying

5.   The proof of main results

With all the results established above, we are adequately prepared for proving Theorem 1.1. First, we state the concept of global weak solution.

Definition 5.1. Let $(n_0, v_0, w_0, u_0)$ satisfy (1.9) and $T\in(0, \infty]$. Then, a fourfold of functions $(n, v, w, u)$ which fulfills

and $n$ as well as $v$ and $w$ are nonnegative in $\Omega \times(0, T)$ and $u$ is divergence-free in $\Omega \times(0, T)$, and

is called a weak solution of problem (1.5) if the following integral identities are satisfied:

for any $\varphi \in C_{0}^{\infty}(\bar{\Omega} \times[0, T))$ satisfying $\frac{\partial \varphi}{\partial \nu} = 0$ on $\partial \Omega \times(0, T)$ and

as well as

for any $\varphi \in C_{0}^{\infty}(\bar{\Omega} \times[0, T))$ and

for any $\varphi \in C_{0}^{\infty}\left(\bar{\Omega} \times[0, T); \mathbb{R}^{2}\right)$ which is divergence-free in $\Omega \times(0, T)$. If $\Omega \times(0, \infty) \longrightarrow \mathbb{R}^{5}$ is a weak solution of (1.5) in $\Omega \times(0, T)$ for all $T > 0$, then $(n, v, w, u)$ is called a global weak solution of (1.5).

In the following auxiliary outcome, we will derive the regularity property of time derivative so as to invoke the Aubin-Lions compactness lemma, which plays a prominent role in proving Theorem 1.1.

Lemma 5.1. If $\alpha > 0$, then for any $T > 0$ and all $\varepsilon \in(0, 1)$, there exists $C(T) > 0$ such that

and

Proof. Firstly, in view of Proposition 4.1, there exists a $C_1 > 0$ such that

Then, testing the first equation in (2.2) by $n_{\varepsilon}$, by virtue of (5.9), we have

Integrating (5.10) over $(0, T)$, (5.7) is valid. Testing the first equation in (2.2) by $\varphi\in C_0^{\infty}(\Omega)$, we conclude there is a $\tilde{C}: = C(C_1, \Omega, C_{S}) > 0$ such that

Therefore, by the definition of the operator norm, one has

Recalling the estimates obtained in (5.7) and (3.37), integrating (5.12) in time, we finally get (5.8).

As an application of the parabolic regularity theory, we may further derive the following Hölder continuity of $n_{\varepsilon}$.

Lemma 5.2. For any $\varepsilon\in(0, 1)$, there exist a positive constant $C$ and $\theta \in (0, 1)$ such that

Proof. Firstly, the first equation of (2.2) can be rewritten as the following sub-problem:

where $a(x, t, \xi): = \xi-n_{\varepsilon}S_{\varepsilon}(x, n_{\varepsilon}, v_{\varepsilon}, w_{\varepsilon}) \nabla v_{\varepsilon}$ and $b(x, t, \xi): = -u_{\varepsilon}\cdot \xi$ with $(x, t, \xi)\in \Omega \times (0, \infty)\times \mathbb{R}^2$. By means of the Young inequality and basic analysis as well as Proposition 4.1, we obtain

and

as well as

with positive constants $C_1$, $C_2$ and $C_3$. Moreover, Proposition 4.1 points out that $|\nabla v_{\varepsilon}|$ and $|\nabla v_{\varepsilon}|^2$ belong to $L^{\infty}((0, \infty); L^p(\Omega))$ for any $p > 1$. In light of the parabolic regularity theory ^[44], for any $\tau > 0$, there exist $\theta: = \theta(\tau) \in(0, 1)$ and some constant $C(\tau) > 0$ such that

which completes the proof. According to classical Schauder estimates, we may exploit the same arguments with Lemmata 9.1, 9.2 and 9.3 in ^[29] to derive the Hölder estimates in $C^{2+\theta, 1+\frac{\theta}{2 }}$ for $v_{\varepsilon}$, $w_{\varepsilon}$ and $u_{\varepsilon}$, so we leave out the details.

Lemma 5.3. If $\alpha > 0$, then for $\tau > 0$, there exist $\theta \in(0, 1)$ and $C(\tau) > 0$ such that the solution of (2.2) satisfies

Based on above preparations, Theorem 1.1 may be proved by utilizing some standard compactness arguments and the parabolic regularity theory.

Lemma 5.4. If $\alpha > 0$, then there exist $\theta \in(0, 1), \{\varepsilon_j\}_{j \in \mathbb{N }} \subset(0, 1)$ and functions

such that $n$, $v$ and $w$ are nonnegative in $\Omega \times(0, T)$, and that

as $\varepsilon = \varepsilon_j \searrow 0$, and $(n, v, w, u, P)$ solves (1.5) classically in $\Omega \times(0, \infty)$.

Proof. By virtue of Proposition 4.1, Lemmata 4.2 and 5.1 and the Arzelà-Ascoli theorem, we can find a sequence $\varepsilon = \varepsilon_j\searrow 0$ as $j\rightarrow \infty$ such that

as well as

and

hold with some limit functions $n, v, w$ and $u$.

By Lemma 5.1, we assert that $n_{\varepsilon}$ belongs to $L^2((0, T); W^{1, 2}(\Omega))$, and $\partial_{t} n_{\varepsilon}$ is bounded in $L^1((0, T); (W_0^{1, 2}(\Omega))^{\ast})$ for any $T > 0$. Noticing the embedding $W^{1, 2}(\Omega)\hookrightarrow\hookrightarrow L^2(\Omega)\hookrightarrow (W_0^{1, 2}(\Omega))^{\ast}$, the Aubin-Lions lemma (^[45]) along with some standard arguments allows us to derive

Now, we may verify the limit functions $n$, $v$, $w$ and $u$ exactly comply with the properties of a weak solution which are stated by Definition 5.1. The integrability conditions in (5.1) and ($\ref{213}$) and the nonnegativity of $n$, $v$ and $w$ are evident by (5.22), (5.23), (5.24), (5.26), (5.27), (5.29), (5.30) and (5.32). Applying the dominated convergence theorem and some standard arguments to the corresponding weak formulations in the regularized problem (2.2) as $\varepsilon = \varepsilon_j\searrow 0$, one can derive the integral identities (5.3)–(5.6) by using (5.22)–(5.32). Moreover, we have

Thus, $(n, v, w, u)$ becomes a global weak solution which exactly enjoys the conditions in Definition 5.1.

Lastly, we claim that this weak solution is virtually a solution in the classical sense. Our method is strongly inspired by Lemma 4.3 in ^[46]. By means of Lemmata 5.2 and 5.3, we obtain

with some $\theta_{1} \in (0, 1)$ and subsequence $\varepsilon = \varepsilon_{j}$. In view of (5.7) and the Hölder regularities provided by (5.34), $n$ possesses the needed regularity properties of a well-established result concerning the gradient Hölder continuity ^[47], which entails

Now, we consider the sub-problem $n_t-\Delta n = g(x, t)$ with boundary condition $\frac{\partial n}{\partial \nu} = h(x, t)\cdot \nu$, where $g: = -\nabla \cdot (nu+nS(x, n, v, w)\nabla v)$ and $h: = nS(x, n, v, w)\nabla v$. As the desired Hölder estimates

and

are warranted by (5.34) and (5.35), invoking the standard parabolic regularity theory ^[48], we can find a $\theta_{3}\in(0, 1)$ such that

This in combination with (5.34) yields a $\theta_{4}\in (0, 1)$ such that

which guarantees the sufficient Hölder regularity of $(n, v, w, u)$ to be a solution in the classical sense and thereby completes the proof.

Finally, Theorem 1.1 is immediate.

Proof of Theorem 1.1. The statement follows from Lemma 5.4 in conjunction with Proposition 4.1.

Acknowledgments

The author would like to express his sincere thanks to anonymous reviewers for their careful reading and constructive suggestions which largely improved this work.

Conflict of interest

The author declares there is no conflict of interest.