1 Introduction

Let Ω be a bounded domain in \({\mathbb {R}^{3}}\) with smooth boundary ∂Ω and ν be the unit outward normal vector to ∂Ω. We consider the regularity problem for the following Keller-Segel-Euler model:

$$\begin{aligned} &\partial_{t}u+u\cdot\nabla u+\nabla\pi+n\nabla\phi=0, \end{aligned}$$
(1.1)
$$\begin{aligned} &\operatorname {div}u=0, \end{aligned}$$
(1.2)
$$\begin{aligned} &\partial_{t}n+u\cdot\nabla n-\Delta n=-\nabla\cdot \bigl(nr(p) \nabla p \bigr), \end{aligned}$$
(1.3)
$$\begin{aligned} &\partial_{t}p+u\cdot\nabla p-\Delta p=-nf(p) \quad\mbox{in } \Omega \times(0,\infty), \end{aligned}$$
(1.4)
$$\begin{aligned} &u\cdot\nu=0, \qquad\frac{\partial n}{\partial\nu}=\frac{\partial p}{\partial\nu}=0 \quad\mbox{on } \partial \Omega \times(0,\infty ), \end{aligned}$$
(1.5)
$$\begin{aligned} &(u,n,p) (\cdot,0)=(u_{0},n_{0},p_{0})\quad \mbox{in } \Omega\subset {\mathbb {R}^{3}}. \end{aligned}$$
(1.6)

Here \(u,\pi,n\) and p denote the fluid velocity field, scalar pressure, cell concentration, and oxygen concentration, respectively. The functions \(f(p)\) and \(r(p)\) are two given smooth functions of p denoting the oxygen consumption rate and chemotactic sensitivity, respectively. The function ϕ denotes the potential function.

When \(\phi=0\), system (1.1) and (1.2) reduces to the well-known Euler system, Ferrari [1] showed the regularity criterion

$$ \operatorname {rot}u\in L^{1} \bigl(0,T;L^{\infty}(\Omega) \bigr). $$
(1.7)

On the other hand, when \(u=0\), system (1.3) and (1.4) reduces to the classical Keller-Segel chemotaxis model [2–4], which received many studies [5–11] on well-posedness and pattern formation of solutions.

For completeness, we also cite [12–14] which show some regularity criteria for the Keller-Segel-Navier-Stokes model.

The aim of this paper is to prove a regularity criterion of local smooth solutions to problem (1.1)-(1.6). We will prove the following.

Theorem 1.1

Let \(u_{0}\in H^{3}, n_{0}, p_{0}\in H^{2}, \operatorname {div}u_{0}=0, n_{0}, p_{0}\geq0\) in Ω and \(n_{0}\cdot\nu=0, \frac{\partial n_{0}}{\partial\nu}=\frac{\partial p_{0}}{\partial\nu}=0\) on ∂Ω. Suppose that ϕ is a smooth function. Let \((u,n,p)\) be a local smooth solution to problem (1.1)-(1.6). If (1.7) and

$$ \nabla p\in L^{\frac{2q}{q-3}} \bigl(0,T;L^{q} \bigr), \quad 3< q\leq \infty, $$
(1.8)

hold true with \(0< T<\infty\), then the solution can be extended beyond \(T>0\).

Remark 1.1

We observe that (1.1)-(1.4) is invariant under the scaling transform \((u,\pi,n, p,\phi)\rightarrow(u_{\lambda},\pi_{\lambda},n_{\lambda},p_{\lambda},\phi_{\lambda})\), where

$$\begin{aligned} &u_{\lambda}:=\lambda u \bigl(\lambda^{2}t,\lambda x \bigr),\qquad \pi_{\lambda}:=\lambda ^{2}\pi \bigl(\lambda^{2}t,\lambda x \bigr), \\ &n_{\lambda}:=\lambda^{2}n \bigl(\lambda^{2}t,\lambda x \bigr),\qquad p_{\lambda}:=p \bigl(\lambda^{2}t,\lambda x \bigr),\qquad \phi_{\lambda}:=\phi \bigl(\lambda^{2}t,\lambda x \bigr). \end{aligned}$$

This implies that the regularity criteria (1.7) and (1.8) are optimal in the sense of scaling.

2 Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. Since local existence results can be proved by using standard arguments, say, Galerkin method, we only deal with the a priori estimates.

First of all, from the equations of \(n,p\) and the maximum principle, we easily see that

$$ n\geq0,0\leq p\leq C,\quad \int n \,dx= \int n_{0} \,dx, $$
(2.1)

where the constant depends only on the initial data.

For any \(m\geq2\), testing (1.3) by \(n^{m-1}\), using the boundary and incompressibility conditions, and denoting \(w:=n^{\frac {m}{2}}\), we calculate

$$\frac{1}{m}\frac{d}{dt} \int w^{2} \,dx+\frac{4(m-1)}{m^{2}} \int \vert \nabla w \vert ^{2} \,dx=(m-1) \int wr(p) (\nabla p\cdot\nabla w)\,dx. $$

Using the smoothness of \(r(p)\) and (2.1), we infer that

$$\begin{aligned} &\frac{1}{m}\frac{d}{dt} \int w^{2} \,dx+\frac{4(m-1)}{m^{2}} \int \vert \nabla w \vert ^{2} \,dx \\ &\quad \leq C \int \vert \nabla p \vert w \vert \nabla w \vert \,dx \\ &\quad\leq C \Vert \nabla p \Vert _{L^{p}} \Vert w \Vert _{L^{\frac{2q}{q-2}}} \Vert \nabla w \Vert _{L^{2}} \\ &\quad\leq C \Vert \nabla p \Vert _{L^{p}} \bigl( \Vert w \Vert _{L^{2}}^{1-\frac{3}{q}} \Vert \nabla w \Vert _{L^{2}}^{1+\frac{3}{q}}+ \Vert w \Vert _{L^{2}} \Vert \nabla w \Vert _{L^{2}} \bigr) \\ &\quad\leq \frac{m-1}{m^{2}} \Vert \nabla w \Vert _{L^{2}}^{2}+C \bigl( \Vert \nabla p \Vert _{L^{p}}^{\frac{2q}{q-3}}+1 \bigr) \Vert w \Vert _{L^{2}}^{2}, \end{aligned}$$

which gives

$$ \Vert n \Vert _{L^{2}(0,T;H^{1})}+ \Vert n \Vert _{L^{\infty}(0,T;L^{m})}\leq C,\quad \forall m\geq 2. $$
(2.2)

Here we have used Young’s inequality and the Gagliardo-Nirenberg inequality for functions on a bounded domain:

$$ \Vert f \Vert _{L^{\frac{2q}{q-2}}}\leq C \bigl( \Vert f \Vert _{L^{2}}^{1-\frac{3}{q}} \Vert \nabla f \Vert _{L^{2}}^{\frac{3}{q}}+ \Vert f \Vert _{L^{2}} \bigr). $$
(2.3)

Testing (1.1) by u, using (1.2) and (2.2), we find that

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int \vert u \vert ^{2} \,dx&= - \int n\nabla\phi\cdot u \,dx \\ &\leq \Vert n \Vert _{L^{3}} \Vert \nabla\phi \Vert _{L^{6}} \Vert u \Vert _{L^{2}} \leq C \Vert \nabla\phi \Vert _{L^{6}} \Vert u \Vert _{L^{2}}, \end{aligned}$$

which gives

$$ \Vert u \Vert _{L^{\infty}(0,T;L^{2})}\leq C. $$
(2.4)

Taking curl to (1.1), using (1.2), we infer that

$$ \partial_{t}\omega+u\cdot\nabla\omega=\omega\cdot\nabla u-\nabla n \times\nabla\phi, $$
(2.5)

where \(\omega:=\operatorname {curl}u\). Testing (2.5) by ω, using (1.2) and (2.2), we deduce

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int \vert \omega \vert ^{2} \,dx&= \int(\omega\cdot\nabla u-\nabla n\times\nabla\phi)\cdot\omega \,dx \\ &\leq \Vert \omega \Vert _{L^{\infty}} \Vert \nabla u \Vert _{L^{2}} \Vert \omega \Vert _{L^{2}}+ \Vert \nabla n \Vert _{L^{2}} \Vert \nabla\phi \Vert _{L^{\infty}} \Vert \omega \Vert _{L^{2}} \\ &\leq C \Vert \omega \Vert _{L^{\infty}} \Vert \omega \Vert _{L^{2}}^{2}+ \Vert \nabla n \Vert _{L^{2}} \Vert \nabla\phi \Vert _{L^{\infty}} \Vert \omega \Vert _{L^{2}}, \end{aligned}$$

which implies

$$\begin{aligned} & \Vert \omega \Vert _{L^{\infty}(0,T;L^{2})}\leq C, \end{aligned}$$
(2.6)
$$\begin{aligned} & \Vert u \Vert _{L^{\infty}(0,T;L^{6})}\leq C. \end{aligned}$$
(2.7)

By using the regularity theory of parabolic equations [15], it follows from (1.3), (1.5), (1.6), (2.1), (2.2), and (2.7) that

$$\begin{aligned} & \Vert \nabla n \Vert _{L^{2}(0,T;L^{\tilde{r}})} \\ &\quad\leq C \bigl(1+ \Vert un \Vert _{L^{2}(0,T;L^{\tilde{r}})}+ \bigl\Vert nr(p)\nabla p \bigr\Vert _{L^{2}(0,T;L^{\tilde{r}})} \bigr) \\ &\quad \leq C \bigl(1+ \Vert u \Vert _{L^{\infty}(0,T;L^{6})} \Vert n \Vert _{L^{\infty}(0,T;L^{\frac {6\tilde{r}}{6-\tilde{r}}})}+ \bigl\Vert r(p) \bigr\Vert _{L^{\infty}} \Vert n \Vert _{L^{\infty}(0,T;L^{\frac{q\tilde{r}}{q-\tilde{r}}})} \Vert \nabla p \Vert _{L^{2}(0,T;L^{\tilde{q}})} \bigr) \\ &\quad \leq C \end{aligned}$$
(2.8)

for some \(3<\tilde{r}<6\) and \(\tilde{r}< q\).

Now we turn to the higher order regularity of the velocity field. Testing (2.5) by \(\vert \omega \vert ^{\tilde{r}-2}\omega\), using (1.2) and (2.8), we obtain

$$\frac{d}{dt} \Vert \omega \Vert _{L^{\tilde{r}}}^{\tilde{r}}\leq C \Vert \omega \Vert _{L^{\infty}} \Vert \omega \Vert _{L^{\tilde{r}}}^{\tilde{r}}+C \Vert \nabla\phi \Vert _{L^{\infty}} \Vert \nabla n \Vert _{L^{\tilde{r}}} \Vert \omega \Vert _{L^{\tilde{r}}}^{\tilde{r}-1}, $$

which gives

$$\begin{aligned} & \Vert \omega \Vert _{L^{\infty}(0,T;L^{\tilde{r}})}\leq C, \end{aligned}$$
(2.9)
$$\begin{aligned} & \Vert u \Vert _{L^{\infty}(0,T;L^{\infty})}\leq C. \end{aligned}$$
(2.10)

Testing (1.1) by \(u_{t}\), using (1.2), (2.2), (2.9), and (2.10), we get

$$\begin{aligned} \Vert u_{t} \Vert _{L^{2}}\leq \Vert u\cdot\nabla u+n \nabla\phi \Vert _{L^{2}} \leq \Vert u \Vert _{L^{\infty}} \Vert \nabla u \Vert _{L^{2}}+ \Vert n \Vert _{L^{3}} \Vert \nabla\phi \Vert _{L^{6}}\leq C, \end{aligned}$$

whence

$$ \Vert u_{t} \Vert _{L^{\infty}(0,T;L^{2})}\leq C. $$
(2.11)

Testing (1.4) by \(-\Delta p\), using (2.1) and (2.2), we deduce

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int \vert \nabla p \vert ^{2} \,dx+ \int \vert \Delta p \vert ^{2} \,dx&= \int \bigl(u\cdot\nabla p-nf(p) \bigr)\Delta p \,dx \\ &\leq \bigl( \Vert u \Vert _{L^{\infty}} \Vert \nabla p \Vert _{L^{2}}+ \bigl\Vert f(p) \bigr\Vert _{L^{\infty}} \Vert n \Vert _{L^{2}} \bigr) \Vert \Delta p \Vert _{L^{2}} \\ &\leq C \bigl( \Vert \nabla p \Vert _{L^{2}}+1 \bigr) \Vert \Delta p \Vert _{L^{2}} \\ &\leq \frac{1}{2} \Vert \Delta p \Vert _{L^{2}}^{2}+C \Vert \nabla p \Vert _{L^{2}}^{2}, \end{aligned}$$

which implies

$$ \Vert p \Vert _{L^{\infty}(0,T;H^{1})}+ \Vert p \Vert _{L^{2}(0,T;H^{2})}\leq C. $$
(2.12)

To achieve higher order regularity of p, we decompose p as

$$p:=p_{1}+p_{2}, $$

where \(p_{1}\) and \(p_{2}\) satisfy

$$\textstyle\begin{cases} \partial_{t}p_{1}-\Delta p_{1}=-\operatorname {div}(up)&\mbox{in } \Omega \times(0,T),\\ \frac{\partial p_{1}}{\partial\nu}=0&\mbox{on }\partial\Omega \times(0,T),\\ p_{1}(x,0)=0&\mbox{in }\Omega \end{cases} $$

and

$$\textstyle\begin{cases} \partial_{t}p_{2}-\Delta p_{2}=-nf(p)& \mbox{in } \Omega\times (0,T),\\ \frac{\partial p_{2}}{\partial\nu}=0&\mbox{on }\partial\Omega \times(0,T),\\ p_{2}(x,0)=p_{0}(x)&\mbox{in }\Omega, \end{cases} $$

respectively.

By using the regularity theory of general parabolic equations (cf. [15]), (2.2), (2.5), and (2.7), we have

$$\begin{aligned} & \Vert \nabla p_{1} \Vert _{L^{m}(0,T;L^{m})}\leq C,\quad \forall m>2, \end{aligned}$$
(2.13)
$$\begin{aligned} &\Vert p_{2} \Vert _{W_{m}^{2,1}(\overline{\Omega}\times[0,T])}\leq C,\quad \forall m>5, \end{aligned}$$
(2.14)

whence

$$ \Vert \nabla p \Vert _{L^{m}(0,T;L^{m})}\leq C. $$
(2.15)

Similarly, by the regularity theory of heat equations [15], we have

$$ \Vert \nabla n \Vert _{L^{m}(0,T;L^{m})}\leq C,\quad \forall m>3. $$
(2.16)

By the well-known \(L^{\infty}\)-estimate of the heat equation, we discover that

$$ \Vert n \Vert _{L^{\infty}(\Omega\times[0,T])}\leq C. $$
(2.17)

Applying \(\partial_{t}\) to (1.3), testing by \(n_{t}\), using (1.2), (2.11), and (2.17), we get

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int n_{t}^{2} \,dx+ \int \vert \nabla n_{t} \vert ^{2} \,dx \\ &\quad= \int u_{t}n\nabla n_{t} \,dx+ \int \bigl(n_{t}r(p)\nabla p+nr'(p)p_{t} \nabla p+nr(p)\nabla p_{t} \bigr)\nabla n_{t} \,dx \\ &\quad\leq \Vert u_{t} \Vert _{L^{2}} \Vert n \Vert _{L^{\infty}} \Vert \nabla n_{t} \Vert _{L^{2}}+C \Vert n_{t} \Vert _{L^{3}} \Vert \nabla p \Vert _{L^{6}} \Vert \nabla n_{t} \Vert _{L^{2}} \\ &\qquad{}+C \Vert n \Vert _{L^{\infty}} \Vert p_{t} \Vert _{L^{3}} \Vert \nabla p \Vert _{L^{6}} \Vert \nabla n_{t} \Vert _{L^{2}}+C \Vert n \Vert _{L^{\infty}} \Vert \nabla p_{t} \Vert _{L^{2}} \Vert \nabla n_{t} \Vert _{L^{2}} \\ &\quad\leq C \Vert \nabla n_{t} \Vert _{L^{2}}+C \Vert n_{t} \Vert _{L^{2}}^{\frac{1}{2}} \Vert \nabla p \Vert _{L^{6}} \Vert \nabla n_{t} \Vert _{L^{2}}^{\frac{3}{2}} \\ &\qquad{}+C \Vert p_{t} \Vert _{L^{3}} \Vert \nabla p \Vert _{L^{6}} \Vert \nabla n_{t} \Vert _{L^{2}}+C \Vert \nabla p_{t} \Vert _{L^{2}} \Vert \nabla n_{t} \Vert _{L^{2}}. \end{aligned}$$
(2.18)

Here we used the fact \(\int n_{t} \,dx=0\) and the Gagliardo-Nirenberg inequality

$$ \Vert n_{t} \Vert _{L^{3}}^{2}\leq C \Vert n_{t} \Vert _{L^{2}} \Vert \nabla n_{t} \Vert _{L^{2}}. $$
(2.19)

Applying \(\partial_{t}\) to (1.4), testing by \(p_{t}\), using (1.2), (2.11), (2.1), and (2.17), we have

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \int p_{t}^{2} \,dx+ \int \vert \nabla p_{t} \vert ^{2} \,dx \\ &\quad= \int u_{t} p\nabla p_{t} \,dx- \int \bigl(n_{t}f(p)+nf'(p)p_{t} \bigr)p_{t} \,dx \\ &\quad\leq \Vert u_{t} \Vert _{L^{2}} \Vert p \Vert _{L^{\infty}} \Vert \nabla p_{t} \Vert _{L^{2}}+C \Vert n_{t} \Vert _{L^{2}} \Vert p_{t} \Vert _{L^{2}}+C \Vert n \Vert _{L^{\infty}} \Vert p_{t} \Vert _{L^{2}}^{2} \\ &\quad\leq C \Vert \nabla p_{t} \Vert _{L^{2}}+C \Vert n_{t} \Vert _{L^{2}} \Vert p_{t} \Vert _{L^{2}}+C \Vert p_{t} \Vert _{L^{2}}^{2}. \end{aligned}$$
(2.20)

Combining (2.18) and (2.20) and using the Gronwall inequality, we conclude that

$$\begin{aligned} & \Vert n_{t} \Vert _{L^{\infty}(0,T;L^{2})}+ \Vert n_{t} \Vert _{L^{2}(0,T;H^{1})}\leq C, \end{aligned}$$
(2.21)
$$\begin{aligned} & \Vert p_{t} \Vert _{L^{\infty}(0,T;L^{2})}+ \Vert p_{t} \Vert _{L^{2}(0,T;H^{1})}\leq C. \end{aligned}$$
(2.22)

Now using the \(H^{2}\)-theory of Poisson’s equation, we have

$$\begin{aligned} & \Vert p \Vert _{L^{\infty}(0,T;H^{2})}+ \Vert p \Vert _{L^{2}(0,T;H^{3})}\leq C, \end{aligned}$$
(2.23)
$$\begin{aligned} & \Vert n \Vert _{L^{\infty}(0,T;H^{2})}+ \Vert n \Vert _{L^{\infty}(0,T;H^{3})}\leq C. \end{aligned}$$
(2.24)

To further improve the regularity of u, we recall some technical lemmas in [1, 16, 17].

Lemma 2.1

[1]

If \(f,g\in H^{s}(\Omega)\cap C(\Omega)\), then

$$ \Vert fg \Vert _{H^{s}}\leq C \bigl( \Vert f \Vert _{H^{s}} \Vert g \Vert _{L^{\infty}}+ \Vert f \Vert _{L^{\infty}} \Vert g \Vert _{H^{s}} \bigr). $$
(2.25)

If \(f\in H^{s}(\Omega)\cap C^{1}(\Omega)\) and \(g\in H^{s-1}(\Omega)\cap C(\Omega)\), then for \(\vert \alpha \vert \leq s\),

$$ \bigl\Vert D^{\alpha}(fg)-fD^{\alpha}g \bigr\Vert _{L^{2}}\leq C \bigl( \Vert f \Vert _{H^{s}} \Vert g \Vert _{L^{\infty}}+ \Vert f \Vert _{W^{1,\infty}} \Vert g \Vert _{H^{s-1}} \bigr). $$
(2.26)

Lemma 2.2

[1, 17]

For any \(u\in H^{3}(\Omega)\) with \(\operatorname {div}u=0\) in Ω and \(u\cdot\nu=0\) on ∂Ω, there holds

$$ \Vert \nabla u \Vert _{L^{\infty}}\leq \bigl(1+ \Vert \operatorname {curl}u \Vert _{L^{\infty}}\log \bigl(e+ \Vert u \Vert _{H^{3}} \bigr) \bigr). $$
(2.27)

Lemma 2.3

[16]

For any \(u\in W^{s,p}\) with \(\operatorname {div}u=0\) in Ω and \(u\cdot\nu=0\) on ∂Ω, there holds

$$ \Vert u \Vert _{W^{s,p}}\leq C \bigl( \Vert u \Vert _{L^{p}}+ \Vert \operatorname {curl}u \Vert _{W^{s-1,p}} \bigr) $$
(2.28)

for any \(s>1\) and \(p\in(1,\infty)\).

Now, applying Δ to (2.5), testing by Δω, using (1.2), (2.25), (2.26), (2.10), (2.28), (2.27), and (2.24), we conclude that

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int \vert \Delta\omega \vert ^{2} \,dx={}&-\sum _{i} \int \bigl[\partial_{i}\Delta(u_{i} \omega)-u_{i}\partial_{i}\Delta\omega \bigr]\cdot \Delta \omega \,dx \\ &{}+ \int\Delta(\omega\cdot\nabla u)\cdot\Delta\omega \,dx- \int \Delta(\nabla n\times\nabla\phi)\cdot\Delta\omega \,dx \\ \leq{}&C \bigl( \Vert \nabla u \Vert _{L^{\infty}} \Vert \Delta\omega \Vert _{L^{2}}+ \Vert \omega \Vert _{L^{\infty}} \Vert \nabla \Delta u \Vert _{L^{2}} \bigr) \Vert \Delta\omega \Vert _{L^{2}} \\ &{}+C \bigl( \Vert \nabla\phi \Vert _{L^{\infty}} \Vert \nabla\Delta n \Vert _{L^{2}}+ \Vert \nabla n \Vert _{L^{\infty}} \Vert \nabla \Delta\phi \Vert _{L^{2}} \bigr) \Vert \Delta\omega \Vert _{L^{2}}, \end{aligned}$$

which gives

$$\begin{aligned} & \Vert \Delta\omega \Vert _{L^{\infty}(0,T;L^{2})}\leq C, \\ & \Vert u \Vert _{L^{\infty}(0,T;H^{3})}\leq C. \end{aligned}$$

This completes the proof of Theorem 1.1.

3 Conclusion

We consider the 3D Keller-Segel-Euler system in a bounded domain. It is a challenging open problem whether the local solution exists globally. Here, a regularity criterion in terms of the vorticity and oxygen concentration is established to guarantee smoothness up to time T. It will help people to gain understanding of the model. We hope to find more inside structures and establish refined regularity criteria.