Abstract
In this paper, we consider the small-convection limit of chemotaxis-Navier–Stokes system with logarithmic sensitivity and logistic-type source
in a bounded convex domain \(\Omega \subseteq \mathbb{R}^{2}\) with smooth boundary, where \(\kappa \in \mathbb{R}\), \(f(s)=\mu _{1} s-\mu _{2} s^{\lambda}\), \(\lambda >1\), and \(\phi :\Omega \rightarrow \mathbb{R}\) is a given smooth potential with second-order partial derivatives. When the chemotaxis sensitivity χ satisfies the appropriate conditions, it is proved that the unique global classical solutions \((n^{\kappa},c^{\kappa},\boldsymbol{u}^{\kappa})\) will stabilize to \((n^{0},c^{0},\boldsymbol{u}^{0})\) as \(\kappa \rightarrow 0\).
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1 Introduction
One of the first mathematical models of chemotaxis was investigated by Keller and Segel [14] to describe the aggregation of certain types of bacteria. In mathematics, it is described as a fully parabolic system
Here, the unknowns \(n=n(t,x)\) and \(c=c(t,x)\) denote the cell density and concentration of chemical, respectively. The physical domain \(\Omega \subset \mathbb{R}^{2}\) is a bounded domain with smooth boundary ∂Ω. The chemotaxis function \(\chi (\cdot )\) denotes the chemotactic sensitivity. In particular, model (1.1) in which chemotactic sensitivity function choices \(\chi (n,c)=\frac{\chi}{c}\) with \(\chi >0\) is an important class of chemotaxis models, its form is suggested by the Weber–Fechner laws and supported by experimental [13] and theoretical evidence [52]. When the chemical reaction is much faster than cell diffusion, system (1.1) could be simplified to parabolic-elliptic equations
this limit process was proved by Wang, Winkler, and Xiang in [36].
In order to study the dynamic behavior of cells under the action of fluid, Tuval et al. [31] took into account the experiment of the collective behavior of Bacillus subtilis in suspension. They observed the formation of plume-like structures and large-scale convection patterns. As an extension of the classical Keller–Segel model, it was used in the case of chemical diffusion and cell migration in a nontrivial interactive fluid environment and was coupled with chemotaxis-fluid equations of the form
Here, \(\boldsymbol{u}=\boldsymbol{u}(t,x)\) and \(P=P(t,x)\) denote the velocity field and the pressure of fluid, respectively, and Ω is a spatial domain where the cells and the fluid interact with each other and move. The given functions \(\chi (n,c)\) and \(g(n,c)\) are the chemotactic sensitivity and the signal function of consumption or production. The potential function ϕ is a scalar-valued function. It can be produced by the different physical mechanism such as gravity, centrifugal. The parameter \(\kappa \in \{0,1\}\) denotes the case of Stokes and Navier–Stokes flows, respectively.
Mathematically, analyzing the above fluid model is very challenging. Tao and Winkler in [28] gave the globally bounded large initial value solution of the problem with Neumann boundary value in a two-dimensional condition for three conditions, whereas for large data up to now only global weak solutions could be established and become eventually smooth and classical. At the same time, there is a globally bounded solution with small initial data. In recent years, there have been many related research works in this regard. For more reference about the chemotaxis-Navier–Stokes system, the corresponding global solvability of classical solutions has been investigated by [6–8, 15, 17, 24, 25, 34, 39, 43, 44, 46, 47, 49, 53] in two- or three-dimensional situation. We also mention complicated variants, e.g., involving rotational flux [4, 5, 18, 19, 22] and logistic source terms [3, 16, 30, 37, 48] as well as nonlinear diffusion [6, 12, 19, 38, 40, 54]. For chemotaxis-Stokes system, the interested reader can refer to earlier works of the global solvability of classical solutions in [1, 20, 27] and nonlinear diffusion in [9, 19, 23, 29, 33, 45] as well as rotational flux in [32, 41, 42].
As for the Navier–Stokes subsystem of (1.2), despite decades of efforts, the global well-posedness still stays in the Dirichlet problem of bounded domain in a two-dimensional case, but the research on the three-dimensional smooth global solution still lacks the viewpoint of mathematical theory. However, the corresponding Stokes case is much more complete [26]. Therefore, some works focus on the chemotaxis-Stokes variant model under a coupled system of the form (1.2). A natural problem is when the chemotactic Navier–Stokes system can approach the chemotactic-Stokes system. The corresponding experimental observations, such as \(Re\approx 10^{-5}\), are in the case of Reynolds number very small. However, rigorous mathematical results are very few. Wang, Winkler, and Xiang [35] and Wu and Xiang [49] gave the mathematical theoretical proof when the solutions of chemotaxis-Navier–Stokes systems will approximate the solution of corresponding chemotaxis-Stokes system in the case of signal consumption and signal generation, respectively.
Under the interaction of chemical signals, a very important coupled system is the chemotaxis model with a logarithmic sensitivity function and a logistic source. Black et al. [2] investigated the model of signal production with logarithmic sensitivity and proved the global existence and uniqueness of classical solutions. Zhao and Zheng in [55] gave the global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type source without fluid. Further, Wu and Natal [51] researched the model in [55] coupled with fluid equations and gave the decay rate of the solutions.
Motivated by the above work, we study the small-convection limit of the following chemotaxis-Navier–Stokes system with logarithmic sensitivity and logistic-type source:
Here, \(\Omega \subset \mathbb{R}^{2}\) is a bounded convex domain with smooth boundary, ν denotes an outer normal vector of ∂Ω, \(f(s)=\mu _{1} s-\mu _{2} s^{\lambda}\), \(\lambda >1\) is a logistic source, the parameters \(\mu _{1}\) and \(\mu _{2}\) are positive constants, the chemotactic sensitivity parameter \(\chi >0\) satisfies
and the initial data satisfy
and the boundary conditions satisfy
For simplicity, we shall assume that \(n_{0} \), \(c_{0}\), \(\boldsymbol{u}_{0}\), ϕ satisfy
where \(A:=-\mathscr{P}\Delta \) denotes the realization of the Stokes operator in \(L^{2}(\Omega ;\mathbb{R}^{2})\), \(D(A):=W^{2,2}(\Omega ;\mathbb{R}^{2})\cap W^{1,2}_{0}(\Omega ; \mathbb{R}^{2})\cap L^{2}_{\sigma}(\Omega )\) denotes the domain and \(L^{2}_{\sigma}(\Omega ):= \{\varphi \in L^{2}(\Omega ;\mathbb{R}^{2}) \mid \nabla \cdot \boldsymbol{u}=0 \}\). The map \(\mathscr{P}: L^{2}(\Omega ;\mathbb{R}^{2})\rightarrow L^{2}_{\sigma}( \Omega )\) is a Helmholtz projection operator.
Theorem 1.1
For all \(\kappa \in {(0, 1)}\), \(p\in (1,\vartheta )\), \(\alpha \in ( \frac{1}{2},1)\), suppose that the initial data \((n_{0},c_{0},\boldsymbol{u}_{0})\) satisfy (1.7) and that \((n^{\kappa},c^{\kappa},\boldsymbol{u}^{\kappa})\) is the unique global solution of system (1.3)–(1.6). Then, for all \(p\in (1,\infty ) \) and any time \(T>0\), there exists a constant \({C(p, T)}>0\) such that
In particular, there is a constant \(C(p,T)>0\) such that
2 Preliminaries
It is necessary for us to give Lemma 2.1, which ensures the global existence and uniqueness of the solutions of our problems.
Lemma 2.1
(Theorem 1.1 in [51])
Let χ satisfy (1.4) and \(\Omega \subset \mathbb{R}^{2}\) be a bounded domain with smooth boundary. If \((n_{0},c_{0},\boldsymbol{u}_{0},\phi )\) fulfils (1.7), then for each \(\kappa \in \mathbb{R}\), system (1.3)–(1.6) admits a unique solution \((n^{\kappa},c^{\kappa},\boldsymbol{u}^{\kappa})\) such that
and that \(n^{\kappa}\) and \(c^{\kappa}\) are positive in \(\Omega \times (0,\infty )\).
Moreover, this solution is uniformly bounded in the sense that
with some positive constant M.
Next, we will use the \(L^{1}\)-mass conservation technique to deal with the logarithmic sensitive function to obtain the uniform lower bound estimation of c, which aims to eliminate the singularity of the logarithmic function.
Lemma 2.2
(Lemma 2.2 and Lemma 2.4 in [51])
For each \(\kappa \in \mathbb{R}\), \(\lambda >1\), it holds that
and
as well as
where \(\theta _{0}\) is a positive constant.
In order to use semigroup estimates, we need the following auxiliary lemma.
Lemma 2.3
(Lemma 3.3 in [10])
Let \(p\in (1,\infty ]\) and let \(\lambda _{1}>0\) denote the first nonzero eigenvalue of −Δ in \(\Omega \subset \mathbb{R}^{N}\). Then there exists \(C>0\) such that, for all \(\varphi \in C^{1}(\bar{\Omega},\mathbb{R}^{N})\) fulfilling \(\varphi \cdot \nu =0\) on ∂Ω, we have
3 Convergence as \(\kappa \rightarrow 0\)
In order to obtain the convergence of \((n^{\kappa},c^{\kappa},\boldsymbol{u}^{\kappa})\rightarrow (n^{0},c^{0},\boldsymbol{u}^{0})\), we need the following transformation.
Let
for each \(\kappa \in (0,1)\), where \(P^{0}\) denotes the pressure of \(\kappa =0\). System (1.3) is transformed into
under the initial conditions
and the boundary conditions
Our key step toward Theorem 1.1 is to derive the corresponding estimate for \((\hat{n},\hat{c},\hat{\boldsymbol{u}})\) with respect to the norm in \((L^{2}(\Omega ))^{4}\). So, we give the following three lemmas.
Lemma 3.1
There is a positive constant C independent of time such that, for any \(\kappa \in (0,1)\), we have
Proof
This proof is based on the standard energy methods. We divide it into five steps.
Step 1. Multiplying equation (3.1)1 by n̂ and integrating by parts, one has
For \(I_{1}\), we can use Hölder’s inequality and Young’s inequality to deduce that
where \(C_{0}>0\) is constant.
For \(I_{2}\), thanks to \(\int _{\Omega}\hat{n}=0\) and using Hölder’s inequality, Young’s inequality, the Gagliardo–Nirenberg inequality, and Poincaré’s inequality, we can infer
where \(C_{1}\), \(C_{2}\), and \({C_{3}:=\max \{2(\mu _{1}+\lambda \mu _{2} M^{ \lambda -1}), 512C_{2}^{4}\|\nabla c^{\kappa}\|^{4}_{L^{4}(\Omega )} \} }\) are positive constants.
For \(I_{3}\), thanks to Lemma 2.2 and using Hölder’s inequality, Young’s inequality, and the Gagliardo–Nirenberg inequality, we find that
with some \(C_{4}>0\) and \(C_{GN}>0\).
For \(I_{4}\), using Hölder’s inequality and Young’s inequality, we see that
with some \(C_{5}>0\).
For \(I_{5}\) and \(I_{6}\), using Lagrange’s mean value theorem and Hölder’s inequality, we deduce that
Substituting \(I_{1}\), \(I_{2}\), \(I_{3}\), \(I_{4}\), \(I_{5}\), \(I_{6}\) into (3.4) and using Lemma 2.1, we have
Step 2. We multiply equation (3.1)2 with ĉ, integrate the resulting equation in Ω, and use the integration by parts to obtain
For \(I_{7}\), we use Hölder’s inequality and Young’s inequality to get
Let \(C_{7}:=\|c^{0}\|^{2}_{L^{\infty}(\Omega )}\). Thus substituting (3.7) into (3.6), we have
Step 3. Testing the second equation of (3.1) with \(-\Delta \hat{c}\) and using the integration by parts, we see that
Using Hölder’s inequality and Young’s inequality, we obtain
Using the Gagliardo–Nirenberg inequality, we have
Substituting (3.11) into (3.10), one has
Thus, we can substitute (3.12) into (3.9) to deduce that
where \(C_{8}:=\max \{2\|{\boldsymbol{u}^{\kappa}}\|^{2}_{L^{\infty}(\Omega )}+1,4C_{GN} \|\nabla c^{0}\|^{2}_{L^{4}(\Omega )} (C_{GN}\|\nabla c^{0}\|^{2}_{L^{4}( \Omega )}+1 ) \}\).
Step 4. Taking the inner product (1.3)3 by \(\hat{\boldsymbol{u}}\), integrating by parts, and using Hölder’s inequality, Poincaré’s inequality, and Young’s inequality, one has
That is,
where \(C_{9}:=\max \{2c^{2}_{1}\|\phi \|^{2}_{L^{\infty}(\Omega )},\| \boldsymbol{u}^{\kappa}\|^{4}_{L^{4}(\Omega )} \}\).
Step 5. Summarily, from Step 1–Step 4, we can close the evolution estimates for n̂, ∇ĉ, and u.
Combining inequalities (3.5), (3.8), (3.13), and (3.14), we obtain
where \(C_{10}:=\max \{2C_{3}+C_{9}+1, 2, 2C_{0}+C_{7}+C_{8} \}\). Then, applying Gronwall’s inequality to (3.15) and \(\hat{n}_{0}=\hat{c}_{0}=\nabla \hat{c}_{0}=0\), \(\hat{\boldsymbol{u}}_{0}=\boldsymbol{0}\), we conclude that Lemma 3.1 holds. □
Lemma 3.2
There exists \(\widetilde{C}>0\) dependent of time such that, for any \(\kappa \in (0,1)\), we find that
and
Proof
Using that \(\alpha <1\) and relying on known regularization properties of the Stokes semigroup in Ω, we obtain \(C_{11}>0\) and \(C_{12}>0\) such that (3.18), where \(f_{1}:=\mathscr{P}[\hat{n}\nabla \phi ]- \kappa \mathscr{P}[ \boldsymbol{u^{\kappa}}\cdot \nabla ]\boldsymbol{u}^{\kappa}\), \(\alpha <1\). Lemma 2.1 and Lemma 3.1 provide
Meanwhile, \(\alpha >\frac{1}{2}\) warrants that \(D(A^{\alpha})\hookrightarrow L^{\infty}(\Omega ;\mathbb{R}^{2})\) [11] holds. □
Lemma 3.3
There is \(\widetilde{C}>0\) such that, for any \(\kappa \in (0,1)\), we see that
Proof
Multiplying equation (3.1)1 by \(\hat{n}^{p-1}\) and integrating by parts, we see that
That is,
For \(J_{1}\), aided by Lemma 2.1 in [50], Hölder’s inequality, Young’s inequality, the Gagliardo–Nirenberg inequality, and Lemma 3.1, we have
For \(J_{2}\), \(J_{3}\), and \(J_{4}\), using Hölder’s inequality, Young’s inequality, the Gagliardo–Nirenberg inequality, and Lemma 3.1 provides some \(c_{i}>0\), \((i=3,\ldots , {10})\) such that
and
as well as
For \(J_{5}\) and \(J_{6}\), using Lagrange’s mean value theorem, Hölder’s inequality, Young’s inequality, and the Gagliardo–Nirenberg inequality, we see that
where \(c_{11}\) and \(c_{12}\) are constants.
Substituting (3.20)–(3.24) into (3.19), we see that there exists a positive constant \(c_{13}\) such that
We apply ∇ to equation (3.1)2 and then multiply the resulting equation by \(|\nabla \hat{c}|^{p-2}\nabla \hat{c}\) to deduce that
Using the pointwise identity \(\Delta |\nabla \hat{c}|^{2}=2\nabla \hat{c}\cdot \nabla \Delta \hat{c}+2|D^{2}\hat{c}|^{2}\), we infer
Since \(\partial _{\nu}\hat{c}=0\) on ∂Ω along with the convexity of Ω ensures that \(\frac{\partial |\nabla c|^{2}}{\partial \nu}\leq 0\) on ∂Ω ([21], Lemma I.1, p.350), we have
For \(K_{2}\), \(K_{3}\), and \(K_{4}\), using Hölder’s inequality, Young’s inequality, the Gagliardo–Nirenberg inequality, and Lemma 3.1 provides some \(c_{i}>0\), \((i=14,\ldots ,19) \) such that
and
as well as
Substituting (3.27)–(3.30) into (3.26), we see that there exists a positive constant \(c_{20}\) such that
Combining with (3.25) and (3.31), there exists \(c_{21}>0\) such that
We may employ the Gagliardo–Nirenberg inequality and Young’s inequality once again to deduce that
where \(C_{15}\), \(C_{16}\), and \(C_{17}\) are positive constants.
Since \(\frac{p-1}{2p}<\frac{p}{4}\), we can combine with (3.32) and (3.33) to see that
Applying Gronwall’s inequality to (3.34), we can complete the proof of Lemma 3.3. □
Lemma 3.4
For any \(\kappa \in (0,1)\), there exists a constant \(\widetilde{C}>0\) such that
Proof
We can rewrite equation (3.1)1 as
where we again abbreviate
Since the initial data \(\hat{n}(x,0)=0\), using the variation-of-constants formula, we get
We can use Lemma 2.3 and Lagrange’s mean value theorem to obtain
In view of Lemma 2.1 and Lemmas 3.2–3.3, from (3.35) we therefore obtain that with some \(c_{23}>0\) we have
as desired. □
Proof of Theorem 1.1
We only need to use Lemmas 3.2–3.4 and combine the embedding \(W^{1,p}(\Omega )\hookrightarrow L^{\infty}(\Omega )\) for \(p>2\) and \(D(A^{\alpha})\hookrightarrow L^{\infty}(\Omega ;\mathbb{R}^{2})\) to complete the proof. □
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Wu, J. Small-convection limit for two-dimensional chemotaxis-Navier–Stokes system with logarithmic sensitivity and logistic-type source. Bound Value Probl 2022, 43 (2022). https://doi.org/10.1186/s13661-022-01622-0
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DOI: https://doi.org/10.1186/s13661-022-01622-0