数理解析研究所講究録
第 1892 巻 2014 年 29-36

29Report on the chemotaxis-fluid systems
Sachiko Ishida
Department of Mathematics
Tokyo University of Science
Abstract. This report deals with the chemotaxis-fluid system (P) with porous-medium
diffusion and considers the uniform-in-time -estimate to solutions $(n, c, u,p)$ to (P),where components of the solution denotes the bacteria density, the chemical concentration,
the velocity field of the fluid and the pressure in the order of inputting. The analysis of
this system is very difficult, however there are few results on the global existence: As tothe two dimensional case, the global existence in (P) with the linear diffusiononand with quasilinear degenerate diffusion $(\Delta n^{m}, m>1)$ on bounded convex domains wasproved. Moreover in the three dimensional case, there are results on global solvability of(P) withonand withon bounded convex domains. As can beseen from these, there are many open interesting problems. This report discusses these
problems and gives the observational result.
$L^{P}$$(\Delta n)$$\triangle n^{\frac{4}{3}}$$\mathbb{R}^{3}$$\mathbb{R}^{2}$$\Delta n^{m}(m>\frac{8}{7})$1. Introduction
The main purpose of our work is to prove the global existence and uniform in timeboundedness in the following chemotaxis-fluid system (KSS) on the $3D$ bounded domain:(KSS)$\{\begin{array}{ll}\frac{\partial n}{\partial t}=\triangle n^{m}-\nabla\cdot(n\nabla c)-u\cdot\nabla n, x\in\Omega, t>0,\frac{\partial c}{\partial t}=\triangle c-nc-u\cdot\nabla c, x\in\Omega, t>0,\frac{\partial u}{\partial t}=\triangle u-\nabla p-n\nabla\phi, x\in\Omega, t>0,\nabla\cdot u=0, x\in\Omega, t>0,\frac{\partial n^{m}}{\partial v}=\frac{\partial n}{\partial v}=0, u=0, x\in\partial\Omega, t>0,n(x, O)=n_{0}(x), c(x, O)=c_{0}(x), u(x, O)=u_{0}(x) , x\in\Omega, t>0,\end{array}$has a smooth boundary
is outward normal derivative on$m>1$ and the initial data are positive functions.

where$\Omega$$\partial\Omega,$$v$$\partial\Omega,$$\phi\in W^{1,\infty}(\Omega)$,

The system (KSS) describes the motion of the swimming bacteria with oxgentaxis
which lives in thin fluid. The bacteria moves toward higher concentration of oxygen andoxygen is consumed by bacteria, and moreover, they are transported by the fluid (cf. Tuval,
et al. [9], Dombrowski, et al. [1] . Here, $(n, c, u,p)$ in (KSS) denotes the bacteria density,
the chemical concentration, the velocity field of the fluid and the pressure, respectively.
$)$

The analysis of this system is difficult because ofin the second equation. When
we consider the -estimate of , gettingin the first equation under control has$u\cdot\nabla c$$L^{p}$$n$$\nabla\cdot(n\nabla c)$

30
a key role. However, in order to estimate this term, we run against the termin thesecond equation which term is associated directly with the third equation. The second
equation looks simple but the estimate ofis hard (at least for the author). In spiteof the difficulty such as this, there are few successful results on the global existence. We
know the following cases have the global solvability:$u\cdot\nabla c$$\nabla c$$\bullet$

$2D$

(a)
(b)
(c)
$\bullet$

$3D$

(a’)
(b)
(c’)
(d’)

case
$\Omega=\mathbb{R}^{2},$$\Omega$$\Omega$$m=1$ , the small initial data (Duan-Lorz-Markowich [2]);is the bounded domain, $m \in(\frac{3}{2},2] (Francesco- Lorz-$Markowich $[3])$ ;is the convex bounded domain, $m>1$ (Tao-Winkler [7]);case$\Omega=\mathbb{R}^{3},$$m=2([3])$ ;$\Omega=\mathbb{R}^{3},$$m= \frac{4}{3}$$\Omega$$\Omega$(Liu-Lorz [6]);is the bounded domain,is the convex bounded domain,$m \in[\frac{7+\sqrt{217}}{12},2]$$m> \frac{8}{7}$([3]);(Tao-Winkler [8]).

As can be seen from these, there are many open interesting problems.

2. Observation and its proof
This report especially considers functions fulfilling the following Cauchy problem(P)$\{\begin{array}{ll}\frac{\partial n}{\partial t}=\Delta(n+\epsilon)^{m}-\nabla\cdot(n\nabla c)-u\cdot\nabla n, x\in \mathbb{R}^{3}, t>0,\frac{\partial c}{\partial t}=\Delta c-nc-u\cdot\nabla c, x\in \mathbb{R}^{3},t>0,\frac{\partial u}{\partial t}=\triangle u-\nabla p-n\nabla\phi, x\in \mathbb{R}^{3}, t>0,\nabla\cdot u=0, x\in \mathbb{R}^{3}, t>0,n(x, 0)=n_{0}(x), c(x, 0)=c_{0}(x), u(x, 0)=u_{0}(x) , x\in \mathbb{R}^{3}, t>0\end{array}$in the classical sense, where
and the initial data
function.
The definition of the classical solution $(n, c, u,p)$ to (P) is as follows:$\epsilon>0$$n_{0},$ $c_{0},$$u_{0}\in C_{0}^{\infty}(\mathbb{R}^{3})$are positive

quadruple $(n, c, u,p)$ is called
Definition 2.1 (classical solutions). Let $T\in(0, \infty)$ .
classical solution to (P) on $[0, T)$ if the following conditions are satisfied:
$A$

$n>0$

and $c>0$and$n\in C^{0}(\mathbb{R}^{3}\cross[o, \tau_{\max}))\cap C^{2,1}(\mathbb{R}^{3}\cross(o, \tau_{\max}))$$c\in C^{0}(\mathbb{R}^{3}\cross[0, T_{\max}))\cap C^{2,1}(\mathbb{R}^{3}\cross(o, \tau_{\max}))$,,$u\in C^{0}(\mathbb{R}^{3}\cross[o, \tau_{\max}))\cap C^{2,1}(\mathbb{R}^{3}\cross(o, \tau_{\max}))$and $(n, c, u,p)$ fulfills the system (P).

$a$

31
Now we present the result.
Proposition 2.1. Let $T>0$ and let $(n, c, u,p)$ be a classical solution to (P) on $[0, T)$ .
Assume that.
$m \in[1, \frac{4}{3})$Then there exists a constant $\delta=\delta(m)>0$ such that if the initial data satisfy the smallness:$\Vert n_{0}\Vert_{L^{1}}, \Vert n_{0}\Vert_{L^{3}}, \Vert n_{0}\log n_{0}\Vert_{L^{1}}, \Vert c_{0}\Vert_{L^{\infty}}, \Vert\nabla c_{0}\Vert_{L^{2}}, \Vert\Delta c_{0}\Vert_{L^{4}}, \Vert u_{0}\Vert_{D(A_{\gamma})}\leq\delta$

where

some

$A_{\gamma}$

is the Stokes operator with dense domain
andclose to 3 and moreover if the additional smallness$D(A_{\gamma})$$\gamma$$:=( \frac{3-p}{4(p+1)}+\frac{2}{3})^{-1}$for$p \in(\frac{3}{2},1)$(Al)$\sup_{0<t<T}(-\int_{\mathbb{R}^{3}}n\log ndx)\leq\delta,$(A2)$\int_{0}^{T}\int_{\mathbb{R}^{3}}|u|^{4}dxd_{S}\leq\delta,$(A3)$\int_{0}^{T}\Vert u\Vert d_{S}\leq\delta\frac{4(p+1)}{L\gamma 3-p}$

are satisfied, then$\sup_{0<t<T}\Vert n(t)\Vert_{L^{p}(\mathbb{R}^{3})}\leq C (\forall p\in[1,3))$

holds, where $C$ does not depend on$\epsilon$

and $T.$

Remark 2.1. We denote the Stokes operator byprojection from, with the domain
onto$A_{p}=-P_{q}\triangle$$(L^{q}(\mathbb{R}^{3}))^{3}$, where$P_{q}$

is the continuous$L_{\sigma}^{q}(\mathbb{R}^{3})$$D(A_{q})=\{w\in L_{\sigma}^{q}(\mathbb{R}^{3});\partial_{i}\partial_{j}w\in(L^{q}(\mathbb{R}^{3}))^{3}(1\leq i,j\leq n)\}.$The spacein$L_{\sigma}^{q}(\mathbb{R}^{3})$is regarded as the closure of$C_{0,\sigma}^{\infty}(\mathbb{R}^{3})$$:=\{w\in(C_{0^{\infty}}(\mathbb{R}^{3}))^{3}, \nabla\cdot w=0\}$$(L^{q}(\mathbb{R}^{3}))^{3}.$2.1. Preliminaries
The first and second lemma. are the maximal Sobolev regularity for Laplace operator$(see e.g., Lady\check{z}enskaja-Solonnikov-Ural’ ceva [5,$ Chapter $IV,$ Section $3])$ and it for Stokes
operator (see e.g., Giga-Sohr [4]).
$3$Lemma 2.2 (Maximal Sobolev regularity for Laplace operator). Let $1<p<\infty,$
and $T>0$ . Then for every $w\in U(O, T;L^{p}(\mathbb{R}^{3}))$ andthere exists a solution
to$N\in \mathbb{N}$$z_{0}\in W^{2,p}(\mathbb{R}^{3})$$z$$z_{t}=\triangle z+w in\mathbb{R}^{3}\cross(0, T) , z(x, O)=z_{0}(x) in\mathbb{R}^{3}$

which satisfies $z\in L^{p}(0, T;W^{2,p}(\mathbb{R}^{3}))$ . Moreover, there exists a positive constantsuch that$C_{\langle p\rangle}(p, N)$$\Vert\triangle z\Vert_{L^{p}(0,T;Lp(\mathbb{R}^{3}))}\leq\Vert\triangle z_{0}\Vert_{L^{p}}+C_{\langle p\rangle}\Vert w\Vert_{Lp(0,T;Lp(\mathbb{R}^{3}))}.$$C_{\langle p\rangle}=$32
Lemma 2.3 (Maximal Sobolev regularity for Stokes operator). Let $1<p,$ $q<\infty,$ $N\in$and$T>0$ and let
be a Stokes operator. Then for every$z_{0}\in D(A_{q})$ there exists a solutionto$w\in L^{q}(0, T;L_{\sigma}^{q}(\mathbb{R}^{3}))$$A_{p}$$\mathbb{N},$$z$$z_{t}=-A_{p}z+w in\mathbb{R}^{3}\cross(0, T) , z(x, O)=z_{0}(x) in\mathbb{R}^{3}$

satisfying $z\in L^{q}(0, T;D(A_{p}))$ . Moreover$z$satisfies the following estimate:$\int_{0}^{T}\Vert A_{p}z(t)\Vert_{L^{p}(\mathbb{R}^{3})}^{q}dt\leq\tilde{C}_{\langle p,q\rangle}(\Vert z_{0}\Vert_{D(A_{p})}^{q}+\int_{0}^{T}\Vert w(t)\Vert_{L^{p}(\mathbb{R}^{3})}^{q}dt)$

where$\tilde{C}_{\langle p,q\rangle}=\tilde{C}_{\langle p,q\rangle}(p, q, N)>0$,

is a constant.

bound forThe next proposition is the mass conservation law for $n(t)$ and the$c(t)$ . These are proved by integration of the first equation and by the parabolic maximum
principle applied to the second equation, respectively.
$L^{\infty}$

Proposition 2.4 ( -conservation law for and-estimate for ). Let $T>0$ and let$(n, c, u,p)$ be a classical solution to (P) on $[0, T)$ . Thensatisfies$L^{1}$$L^{\infty}$

$n$

$c$

$n$

and

$c$

$\Vert n(t)\Vert_{L^{1}(\mathbb{R}^{3})}=\Vert n_{0}\Vert_{L^{1}}$for all $t\in(O, T)$

for all$(x, t)\in \mathbb{R}^{3}\cross(0, T)$satisfies$|c|\leq\Vert c_{0}\Vert_{L}\infty$.

Lemma 2.5 is the energy estimate for (P). The proof is came from [8, Lemma 2.3].
Lemma 2.5 (Energy estimate). Let $T>0$ and let $(n, c, u,p)$ be a classical solution to(P) on $[0, T)$ . Then there exists a positive constantsuch that for any $t\in(0, T)$$k_{1}$$\frac{d}{dt}\{\int_{\mathbb{R}^{3}}n\log n+2\int_{\mathbb{R}^{3}}|\nabla\sqrt{c}|^{2}\}+\int_{\mathbb{R}^{3}}|\nabla n^{\frac{m}{2}}|^{2}+\int_{\mathbb{R}^{3}}c|D^{2}\log c|^{2}+\frac{1}{2}\int_{\mathbb{R}^{3}}n\frac{|\nabla c|^{2}}{c}\leq k_{1}\int_{\mathbb{R}^{3}}|u|^{4}.$The following estimate is the estimate to
2.3, 2.5] and the above energy estimate.

$\nabla c$. The proof is also came from [8, Lemmas). Let $T>0$ and let $(n, c, u,p)$ be a classical solution to
Lemma 2.6 ( -estimate to(P) on $[0, T)$ . Then there exists a positive constantsuch that$L^{4}$$\nabla c$$K_{1}$$\int_{0}^{t}\int_{\mathbb{R}^{3}}|\nabla c|^{4}dxds<K_{1}, (\forall t\in(0, T))$

where

$K_{1}$

depends on the initial data and$\sup_{0<s<t}(-\int_{\mathbb{R}^{3}}n\log ndx),$$\int_{0}^{t}\int_{\mathbb{R}^{3}}|u|^{4}dxds.$

Remark 2.2. As to Lemma 2.6, the assumption of smallness means thatbe to be small.

$\int_{0}^{t}\int_{\mathbb{R}^{3}}|\nabla c|^{4}$

can

33
2.2. Proof of main proposition
In this subsection, we compute the $U$ estimate to $n(t)(\forall p\in[1,3))$ . For simplicity,
we discuss (P) withthroughout the following. Let us start computing.
Let $p>1.$
the straightforward argument we have$\epsilon=0$$\mathbb{R}om$$\frac{1}{p}\frac{d}{dt}\Vert n(t)\Vert_{L^{p}(Jl)}^{p}=-\int_{tl}\nabla n^{m}\cdot\nabla n^{p-1}+\int_{tl}n\nabla c\cdot\nabla n^{p-1}+\int_{11}u\nabla nn^{p-1}$$=- \frac{4m(p-1)}{(m+p-1)^{2}}\int_{l}|\nabla n^{\underline{m+}g\underline{-1}}2|^{2}+(p-1)\int_{l}\nabla c\cdot n^{p-1}\nabla n$$=- \frac{2m(p-1)}{(m+p-1)^{2}}\int_{\Omega}|n^{\underline{m}\ovalbox{\tt\small REJECT}+\underline{-1}}|^{2}+\frac{(p-1)}{p}\int_{\Omega}\nabla c\nabla n^{p}$$\leq-\frac{2m(p-1)}{(m+p-1)^{2}}\Vert\nabla n^{\underline{m}\mu_{2}\underline{-1}}+\Vert_{L^{2}(tl)}^{2}+\frac{(p-1)}{p}\int_{l}|\triangle c|n^{p}.$Integrating the above estimate over $(0, t)$ , we obtain from the Young inequality withand Lemma 2.2 that$( \frac{p}{p+1},\frac{1}{p+1})$$\Vert n(t)\Vert_{L^{p}(\Omega)}^{p}-\Vert n_{0}\Vert_{L^{p}(\Omega)}^{p}$$\leq-\frac{2mp(p-1)}{(m+p-1)^{2}}\int_{0}^{t}\Vert\nabla n^{\underline{m}p_{2}\underline{-1}}+\Vert_{L^{2}(t1)}^{2}$$+(p-1) \int_{0}^{t}\int_{l}|\triangle c|^{p+1}+(p-1)\int_{0}^{t}\int_{\iota}|n|^{p+1}$$\leq-\frac{2mp(p-1)}{(m+p-1)^{2}}\int_{0}^{t}\Vert\nabla n^{\frac{m+\mathcal{D}-1}{2}}\Vert_{L^{2}(t1)}^{2}$$+c_{1}(p-1) \{\Vert\Delta c_{0}\Vert_{L^{p+1}}+\int_{0}^{t}\int_{l}|-nc-u\cdot\nabla c|^{p+1}\}+(p-1)\int_{0}^{t}\int_{tl}|n|^{p+1}$

with some positive constant
Proposition 2.4 ensures that(2.1)

$c_{1}$. Hence the estimate$\Vert c(t)\Vert_{L}\infty\leq\Vert c_{0}\Vert_{L}\infty(t>0)$

from$\Vert n(t)\Vert_{L^{p}(\mathfrak{l}l)}^{p}-\Vert n_{0}\Vert_{L^{p}(J1)}^{p}$$\leq-\frac{2mp(p-1)}{(m+p-1)^{2}}\int_{0}^{t}\Vert\nabla n^{\underline{m}+p_{2}\underline{-1}}\Vert_{L^{2}(t1)}^{2}+c_{1}(p-1)\Vert\triangle c_{0}\Vert_{L^{p+1}}$$+(p-1)(c_{1} \Vert c_{0}\Vert_{L}\infty+1)\int_{0}^{t}\int_{f}|n|^{p+1}+c_{1}(p-1)\int_{0}^{t}\int_{l}|u\cdot\nabla c|^{p+1}.$

Firstly, we consider the third term in the right hand side of (2.1). The Gagliardo-Nirenberg
inequality yields for any $p> \frac{m}{2}+1$$\int_{0}^{t}\int_{tl}|n|^{p+1}=\int_{0}^{t}\Vert n^{e_{\frac{+m-1}{2}\Vert}\frac{2(p+1)}{L^{p}p+m-1*_{m-}^{2+1}}}\prec$$\leq c_{2}\int_{0}^{t}(\frac{+m-1}{2}$34with some constant $c_{2}>0$ and$\alpha:=\frac{p+m-1}{2}(\frac{2}{3(p-1)}-\frac{1}{p+1})(\frac{p+m-1}{6(2-m)}-\frac{1}{6})$Note that $\alpha\in(0,1)$ when $m<2$ . Because of(2.2)$\frac{2\alpha(p+1)}{p+m-1}=2$.

, we have$\int_{0}^{t}\int_{\zeta\}}|n|^{p+1}dxds\leq c_{2}\int_{0}^{t}\Vert\nabla n\frac{p+m-1}{2}\Vert_{L^{2}}^{2}\Vert n\Vert_{L^{\lrcorner 3}}^{2-mfi^{-\lrcorner 1}}ds.$

Next, $we’ d$ like to estimate the last term in the right hand side of (2.1). The Younginequality withgives for any $p<3$$(E_{\frac{+1}{4}},4-(p4+1))$(2.3)$\int_{0}^{t}\int_{\{\}}|u\cdot\nabla c|^{p+1}\leq\int_{0}^{t}\int_{1}|\nabla c|^{4}+\int_{0}^{t}\int_{tl}|u|^{\frac{4(p+1)}{4-(p+1)}}.$In order to estimate the second term in the right hand side of (2.3), we use the embedding
and Lemma 2.3. Then we see that$W^{2,\gamma}\hookrightarrow L^{-}-p^{\frac{+4}{+3}}4p$$( \gamma :=(\frac{-p+3}{4p+4}+\frac{2}{3})^{-1})$$\int_{0}^{t}\int_{\zeta\}}|u|-p^{\frac{+4}{+3}}\underline{4}_{R}\leq c_{3}\int_{0}^{t_{4\Leftrightarrow 4}}\Vert u\Vert_{W^{2,\gamma}}^{-\nu+3}-B$$\leq c_{3}\int_{0}^{t_{4}}\Vert u\Vert_{L^{\gamma}}^{-p^{\frac{+4}{+3}}}+c_{3}-l\int_{0}^{t_{\underline{4}_{L}}}\Vert\triangle u\Vert_{L^{\gamma}}^{-p^{\frac{+4}{+3}}}$$\leq c_{3}\int_{0}^{t}\frac{4}{L}p+3\int_{0}t_{4}\Vert-p+$$\leq c_{3}\int_{0}^{t_{4+}}\Vert u\Vert_{L^{\gamma}}^{-p+}+c_{4}\Vert u_{0}\Vert_{D(A_{\gamma})}+c_{4}\Vert\nabla\phi\Vert_{L}\infty-arrow\frac{4}{3}\int_{0}^{t_{4}}\Vert n(s)\Vert_{L^{\gamma}}^{-p^{\frac{+4}{+3}}}-A$, wheredue toare positive constants. In light of the Gagliardo-Nirenberg
inequality we can find a positive constant such that$\phi\in W^{1,\infty}$$c_{3},$$c_{4}$$c_{5}$(2.4)$\int_{0}^{t}\int_{\{\}}|u|^{-\Leftrightarrow\frac{4}{3}}-p+4+\leq K_{2}+c_{4}\Vert u_{0}\Vert_{D(A_{\gamma})}+c_{5}\Vert\nabla\phi\Vert_{L}\infty\int_{0}^{t}\Vert\nabla n^{L_{\frac{+m-1}{2}}}(s)\Vert_{L^{2}}^{2}\Vert n(t)\Vert_{L^{q_{1}}}^{\alpha}2,$$-4L\underline{+4}$$:=3 \{\frac{3-p}{4p+4}-(p+m-1)\}\{1+\frac{3-p}{2(p+1)}\}^{-1}$ and somewhere
Hence connecting (2.4) into (2.3), we see that for any $p \in[\frac{m}{2}+1,3)$$K_{2}$(2.5)$:=c_{3} \int_{0}^{t}\Vert u\Vert_{L^{\gamma}}^{-p+3}ds,$$q_{1}$$\alpha_{2}>0.$$\int_{0}^{t}\int_{\zeta\}}|u\cdot\nabla c|^{p+1}$$\leq K_{1}+K_{2}+\Vert u_{0}\Vert_{D(A_{\gamma})}+\Vert\nabla\phi\Vert_{L^{\infty}}\int_{0}^{t}2,$where $K_{1}>0$ is the same constant as in Lemma 2.6. Then the estimates (2.1), (2.2) and(2.5) provide

35$\Vert n(t)\Vert_{L^{p}(l)}^{p}-\Vert n_{0}\Vert_{L^{p}(tl)}^{p}$$\leq-\frac{2mp(p-1)}{(m+p-1)^{2}}\int_{0}^{t}\Vert\nabla n^{\underline{m+}\epsilon\underline{-1}}2\Vert_{L^{2}}^{2}+c_{1}(p-1)\Vert\Delta c_{0}\Vert_{L^{p+1}}$$+(p-1)(c_{1} \Vert c_{0}\Vert_{L}\infty+1)\int_{0^{L^{\Delta\mapsto-1}}}^{t}\Vert\nabla n^{e_{\frac{+m-1}{2}}}\Vert_{L^{2}}^{2}\Vert n\Vert_{3}^{2-m}ds$$+(p-1) \{K_{1}+K_{2}+c_{4}\Vert u_{0}\Vert_{D(A_{\gamma})}+c_{5}\Vert\nabla\phi\Vert_{L^{\infty}}\int_{0}^{t}\Vert\nabla n^{\epsilon_{\frac{+m-1}{2}}}(s)\Vert_{L^{2}}^{2}\Vert n(t)\Vert_{L^{q_{1}}}^{\alpha 2}ds\}.$Then the smallness of the initial data gives uniform-in-time $U$ -estimate on
for small$t_{1}>0$ and for any $p \in[\frac{m}{2}+1,3)$ . The mass conservation law with this boundedness entails
uniform-in-time -estimate on
for any $p\in[1,3)$ . Moreover, repeating the same
argument for the sake of a priori estimate and the assumption (Al), (A2), (A3) finallyensures -estimate on $[0, T)$ , and therefore this completes the $pro$ of.
$[0, t_{1})$$L^{p}$$[0, t_{1})$$L^{p}$

Remark 2.3. Our main purpose of this work is to find the global solvability in (KSS)
especially with. Since this report could not achieve it the author would like toresearch it in the future.
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Department of Mathematics
Tokyo University of Science
1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, JAPANE–mail address: s-ishida@rs.tus.ac.jp$E_{R}^{rightarrow}Efl\lambda^{\frac{\backslash }{\mp}}\cdot g_{\mp}^{\simeq_{p}^{\backslash }r\ovalbox{\tt\small REJECT}_{p}}-*\mathscr{Z}R^{\backslash }\mp r\}fiffl\dagger\yen\neq$