Global well-posedness for the 3D incompressible Keller–Segel–Navier–Stokes equations

Abstract

The Keller–Segel–Navier–Stokes system

$$\begin{aligned} \left\{ \begin{aligned}&\rho _{t}+u\cdot \nabla \rho =\Delta \rho -\nabla \cdot (\rho \nabla c)-\rho ^2,\\&c_{t}+u\cdot \nabla c=\Delta c -c+\rho ,\\&u_{t}+u\cdot \nabla u+\nabla P=\Delta u-\rho \nabla \phi ,\\&\nabla \cdot u=0, \end{aligned} \right. \end{aligned}$$

is considered in $\mathbb R^3$. It is proved that under the assumptions that

$$\begin{aligned} \left( \Vert u^h_0\Vert _{B_{p,1}^{-1+\frac{3}{p}}}+\Vert \rho _0\Vert _{B_{q,1}^{-2 +\frac{3}{q}}}+\Vert c_0\Vert _{B_{q,1}^{\frac{3}{q}}}\right) \mathrm{exp}\left\{ C_0\left( 1+\Vert u^3_0\Vert _{B_{p,1}^{-1+\frac{3}{p}}}\right) ^2\right\} \le \gamma , \end{aligned}$$

we obtain the global well-posedness for the three-dimensional incompressible Keller–Segel–Navier–Stokes equations with large initial vertical velocity component by using the Fourier frequency localization and Bony paraproduct decomposition.

Appendix

In this “Appendix,” we will give a sketch proof of local existence and how to find the maximal time $T_{\max }$.

Lemma 6.1

Let $(X,\Vert \cdot \Vert )$ be a Banach space, $B:X\times X\rightarrow X$ a bilinear operator with norm K and $L:X\rightarrow X$ a continuous operator with norm $M<1$. Let $y\in X$ satisfy $4K\Vert y\Vert _X<(1-M)^2$. Then, the equation $u=y+L(u)+B(u)$ has a unique solution in the ball $B(0,{1-M\over 2K})$.

Proposition 6.1

Let $u_L=\text {e}^{t\Delta }u_0$, $\rho _L=\text {e}^{t\Delta }\rho _0$ and $c_L=\text {e}^{t\Delta }c_0$. $(u,\ \rho ,\ c)$ is a mild solution of system (1.7) on $[0,T]\times \mathbb R^3$ with initial data $(u_0,\ \rho _0,\ c_0)$ if and only if $(u,\ \rho ,\ c,)=(u_L+{{\bar{u}}},\ \rho _L+{\bar{\rho }},\ c_L+{{\bar{c}}})$ with

$$\begin{aligned} {{\bar{u}}}= & {} -\int \limits _0^t\text {e}^{(t-s)\Delta }{{\mathcal {P}}}(u_L\cdot \nabla u_L+\rho _L\nabla \phi )(\cdot ,s)\text {d}s\\&-\int \limits _0^t\text {e}^{(t-s)\Delta }{{\mathcal {P}}}(u_L\cdot \nabla {{\bar{u}}}+\bar{u}\cdot \nabla u_L+{\bar{\rho }}_L\nabla \phi )(\cdot ,s)\text {d}s\\&-\int \limits _0^t\text {e}^{(t-s)\Delta }{{\mathcal {P}}}({{\bar{u}}}\cdot \nabla \bar{u})(\cdot ,s)\text {d}s\\= & {} y_1+L_1({\bar{\rho }},{{\bar{c}}},{{\bar{u}}})+B_1(({\bar{\rho }},{{\bar{c}}},\bar{u}),({\bar{\rho }},{{\bar{c}}},{{\bar{u}}})),\\ \bar{\rho }= & {} -\int \limits _0^t\text {e}^{(t-s)\Delta }(u_L\cdot \nabla \rho _L+\nabla \cdot (\rho _L\nabla c_L)+\rho _L\rho _L)(\cdot ,s)\text {d}s\\&-\int \limits _0^t\text {e}^{(t-s)\Delta }(u_L\cdot \nabla {\bar{\rho }}+{{\bar{u}}}\cdot \nabla \rho _L+\nabla \cdot (\rho _L\nabla {{\bar{c}}})+\nabla \cdot ({\bar{\rho }}\nabla c_L)+{\bar{\rho }}\rho _L+\rho _L{\bar{\rho }})(\cdot ,s)\text {d}s\\&-\int \limits _0^t\text {e}^{(t-s)\Delta }(\bar{u}\cdot \nabla {\bar{\rho }}+\nabla \cdot ({\bar{\rho }}\nabla \bar{c})+{\bar{\rho }}{\bar{\rho }})(\cdot ,s)\text {d}s\\= & {} y_2+L_2({\bar{\rho }},{{\bar{c}}},{{\bar{u}}})+B_2(({\bar{\rho }},{{\bar{c}}},\bar{u}),({\bar{\rho }},{{\bar{c}}},{{\bar{u}}})), \end{aligned}$$

and

$$\begin{aligned} {{\bar{c}}}= & {} -\int \limits _0^t\text {e}^{(t-s)\Delta }(u_L\cdot \nabla c_L+c_L-\rho _L)(\cdot ,s)\text {d}s -\int \limits _0^t\text {e}^{(t-s)\Delta }(u_L\cdot \nabla {{\bar{c}}}+{{\bar{u}}}\cdot \nabla c_L)(\cdot ,s)\text {d}s\\&-\int \limits _0^t\text {e}^{(t-s)\Delta }({{\bar{u}}}\cdot \nabla {{\bar{c}}}+\bar{c}-{\bar{\rho }})(\cdot ,s)\text {d}s =y_3+L_3({\bar{\rho }},{{\bar{c}}},{{\bar{u}}})+B_3(({\bar{\rho }},{{\bar{c}}},\bar{u}),({\bar{\rho }},{{\bar{c}}},{{\bar{u}}})) \end{aligned}$$

with

$$\begin{aligned} ({{\bar{u}}},{\bar{\rho }},{{\bar{c}}})|_{t=0}=(0,0,0). \end{aligned}$$

Proof

We skip the proof because it is easy. $\square$

The proof of local existence is standard, and one can refer to [36]. For convenience to readers, we give a sketch. Applying Lemma 2.2 , we notice

$$\begin{aligned}&\Vert u_L\Vert _{{{\widetilde{L}}}^\infty ([0,T];B_{p,1}^{-1+{3\over p}})}\le C\Vert u_0\Vert _{B_{p,1}^{-1+{3\over p}}},\\&\Vert \rho _L\Vert _{{{\widetilde{L}}}^\infty ([0,T];B_{q,1}^{-2+{3\over q}})}\le C\Vert \rho _0\Vert _{B_{q,1}^{-2+{3\over q}}},\\&\Vert c_L\Vert _{{{\widetilde{L}}}^\infty ([0,T];B_{q,1}^{{3\over q}})}\le C\Vert c_0\Vert _{B_{q,1}^{{3\over q}}}, \end{aligned}$$

and

$$\begin{aligned}&\lim _{T\rightarrow 0}\Vert u_L\Vert _{L^1 ([0,T];B_{p,1}^{1+{3\over p}})}=0,\\&\lim _{T\rightarrow 0}\Vert \rho _L\Vert _{L^1 ([0,T];B_{q,1}^{3\over q})}=0,\\&\lim _{T\rightarrow 0}\Vert c_L\Vert _{L^1 ([0,T];B_{q,1}^{2+{3\over q}})}=0.\\ \end{aligned}$$

Let

$$\begin{aligned}&T_1=\sup \{T>0:\Vert u_L\Vert _{L^1 ([0,T];B_{p,1}^{1+{3\over p}})}\le \eta \},\\&T_2=\sup \{T>0:\Vert \rho _L\Vert _{L^1 ([0,T];B_{q,1}^{3\over q})}\le \eta \},\\&T_3=\sup \{T>0:\Vert c_L\Vert _{L^1 ([0,T];B_{q,1}^{2+{3\over q}})}\le \eta \}. \end{aligned}$$

Choose ${{\mathcal {T}}}=\min \{T_1,T_2,T_3\}$ and take $T\le {\mathcal T}$ and $\eta$ small enough. Set

$$\begin{aligned} {{\mathcal {F}}}_T=L^1 ([0,T];B_{p,1}^{-1+{3\over p}})\times L^1 ([0,T];B_{q,1}^{-2+{3\over q}})\times L^1 ([0,T];B_{q,1}^{{3\over q}}). \end{aligned}$$

By Lemma 4.1–4.3, there exists a constant $K>0$ such that

$$\begin{aligned} \Vert B\Vert _{\Theta _T}= & {} \Vert (B_1,B_2,B_3)(({{\bar{u}}},{\bar{\rho }},{{\bar{c}}}),(\bar{u},{\bar{\rho }},{{\bar{c}}}))\Vert _{\Theta _T}\\\le & {} C\Vert {{\bar{u}}}\cdot \nabla {{\bar{u}}}+\bar{u}\cdot \nabla {\bar{\rho }}+\nabla \cdot ({\bar{\rho }}\nabla {{\bar{c}}})+{\bar{\rho }}{\bar{\rho }}+\bar{u}\cdot \nabla {{\bar{c}}}+{{\bar{c}}}-{\bar{\rho }}\Vert _{{{\mathcal {F}}}_T}\\\le & {} K\Vert ({{\bar{u}}},{\bar{\rho }},{{\bar{c}}})\Vert ^2_{\Theta _T}. \end{aligned}$$

On the other hand, we have

$$\begin{aligned}&\Vert L\Vert _{\Theta _T}=\Vert (L_1,L_2,L_3)({{\bar{u}}},{\bar{\rho }},\bar{c}\Vert _{\Theta _T}\\&\quad \le C\Vert u_L\cdot \nabla {{\bar{u}}}+{{\bar{u}}}\cdot \nabla u_L+{\bar{\rho }}_L\nabla \phi +u_L\cdot \nabla {\bar{\rho }}+{{\bar{u}}}\cdot \nabla \rho _L+\nabla \cdot (\rho _L\nabla {{\bar{c}}})\\&\qquad +\,\nabla \cdot ({\bar{\rho }}\nabla c_L)+{\bar{\rho }}\rho _L+\rho _L{\bar{\rho }}+u_L\cdot \nabla {{\bar{c}}}+{{\bar{u}}}\cdot \nabla c_L\Vert _{{{\mathcal {F}}}_T}\\&\quad \le M\Vert ({{\bar{u}}},{\bar{\rho }},{{\bar{c}}}\Vert _{\Theta _T}, \end{aligned}$$

where we can choose $M<1$ if $\eta$ is sufficiently small.

According to estimates for heat equation (see [18]), for $E_0=B_{p,1}^{-1+{3\over p}}\times B_{q,1}^{-2+{3\over q}}\times B_{q,1}^{{3\over q}}$, we obtain

$$\begin{aligned} \Vert y\Vert _{\Theta _T}= & {} \Vert (y_1,y_2,y_3)\Vert _{\Theta _T}\\\le & {} C\Vert u_L\cdot \nabla u_L+\rho _L\nabla \phi +u_L\cdot \nabla \rho _L+\nabla \cdot (\rho _L\nabla c_L)+\rho _L\rho _L+u_L\cdot \nabla c_L+c_L-\rho _L\Vert _{{{\mathcal {F}}}_T}\\\le & {} C\eta \Vert (u_0,\rho _0,c_0\Vert _{E_0}. \end{aligned}$$

We can take $\eta$ and T small enough such that $C\eta \Vert (u_0,\rho _0,c_0)\Vert _{E_0}\le {(1-M)^2\over 4K}$. By Lemma 6.1 and Proposition 6.1, we have a unique local solution of (1.7).

Next, we prove the solution can be extended from $I=[0,T]$ to $I_1=[T,T_1]$ for some $T_1>T$. Consider the following equalities:

$$\begin{aligned}&u(t)=\text {e}^{(t-T)\Delta }u(T)-\int \limits _T^t\text {e}^{(t-s)}{{\mathcal {P}}}(u\cdot \nabla u+\rho \nabla \phi )(\cdot ,s)\text {d}s,\\&\rho (t)=\text {e}^{(t-T)\Delta }\rho (T)-\int \limits _T^t\text {e}^{(t-s)}(u\cdot \nabla \rho +\nabla \cdot (\rho \nabla c)+\rho ^2)(\cdot ,s)\text {d}s,\\&c(t)=\text {e}^{(t-T)\Delta }c(T)-\int \limits _T^t\text {e}^{(t-s)}(u\cdot \nabla c+c-\rho )(\cdot ,s)\text {d}s. \end{aligned}$$

By the same line in proving the local existence above, together with the Banach’s fixed point Lemma 6.1, we obtain the local solution on $[T,T_1]$. Then, we can find a maximal $T_{\max }>0$ step by step.