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A regularity criterion for the Keller-Segel-Euler system
Boundary Value Problems volume 2017, Article number: 124 (2017)
Abstract
We consider a Keller-Segel-Euler model and prove a regularity criterion of the local strong solutions in a 3D bounded domain Ω.
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:
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
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
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
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
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
Using the smoothness of \(r(p)\) and (2.1), we infer that
which gives
Here we have used Young’s inequality and the Gagliardo-Nirenberg inequality for functions on a bounded domain:
Testing (1.1) by u, using (1.2) and (2.2), we find that
which gives
Taking curl to (1.1), using (1.2), we infer that
where \(\omega:=\operatorname {curl}u\). Testing (2.5) by ω, using (1.2) and (2.2), we deduce
which implies
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
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
which gives
Testing (1.1) by \(u_{t}\), using (1.2), (2.2), (2.9), and (2.10), we get
whence
Testing (1.4) by \(-\Delta p\), using (2.1) and (2.2), we deduce
which implies
To achieve higher order regularity of p, we decompose p as
where \(p_{1}\) and \(p_{2}\) satisfy
and
respectively.
By using the regularity theory of general parabolic equations (cf. [15]), (2.2), (2.5), and (2.7), we have
whence
Similarly, by the regularity theory of heat equations [15], we have
By the well-known \(L^{\infty}\)-estimate of the heat equation, we discover that
Applying \(\partial_{t}\) to (1.3), testing by \(n_{t}\), using (1.2), (2.11), and (2.17), we get
Here we used the fact \(\int n_{t} \,dx=0\) and the Gagliardo-Nirenberg inequality
Applying \(\partial_{t}\) to (1.4), testing by \(p_{t}\), using (1.2), (2.11), (2.1), and (2.17), we have
Combining (2.18) and (2.20) and using the Gronwall inequality, we conclude that
Now using the \(H^{2}\)-theory of Poisson’s equation, we have
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
If \(f\in H^{s}(\Omega)\cap C^{1}(\Omega)\) and \(g\in H^{s-1}(\Omega)\cap C(\Omega)\), then for \(\vert \alpha \vert \leq s\),
Lemma 2.2
For any \(u\in H^{3}(\Omega)\) with \(\operatorname {div}u=0\) in Ω and \(u\cdot\nu=0\) on ∂Ω, there holds
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
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
which gives
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.
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Acknowledgements
The first, second and fourth authors are partially supported by the National Natural Science Foundation of China (Grant No. 11171154 and 11501346). The third author extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).
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Fan, J., Liu, D., Samet, B. et al. A regularity criterion for the Keller-Segel-Euler system. Bound Value Probl 2017, 124 (2017). https://doi.org/10.1186/s13661-017-0860-3
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DOI: https://doi.org/10.1186/s13661-017-0860-3