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Local well-posedness for the incompressible full magneto-micropolar system with vacuum

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Abstract

In this paper, a full incompressible magneto-micropolar system is investigated. We prove the local well-posedness of strong solutions to the full system with vacuum. Moreover, previous compatibility conditions on the initial data are also moved.

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Acknowledgements

This work is partially supported by NSFC (No. 11971234).

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Correspondence to Yong Zhou.

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Appendix: Proof of (1.18)

Appendix: Proof of (1.18)

Proof

Let \(w\times n=0\) on \(\partial \varOmega \). If (1.18) is not true, there exists a sequence \(w_m\in H^1\) and \(w_m\times n=0\) on \(\partial \varOmega \), such that

$$\begin{aligned} \Vert w_m\Vert _{L^2}>m(\Vert \mathrm {div}\,w_m\Vert _{L^2}+\Vert \mathrm {rot}\,w_m\Vert _{L^2}),\ \ \forall m. \end{aligned}$$
(2.33)

We may assume that

$$\begin{aligned} \Vert w_m\Vert _{L^2}=1. \end{aligned}$$
(2.34)

Using (1.17) for \(s=1\), we have

$$\begin{aligned} \Vert w_m\Vert\le & {} C(\Vert \mathrm {div}\,w_m\Vert _{L^2}+\Vert \mathrm {rot}\,w_m\Vert _{L^2}+\Vert w_m\Vert _{L^2})\nonumber \\\le & {} C\left( \frac{1}{m}+1\right) \le 2C. \end{aligned}$$
(2.35)

We can extract a subsequence still denoted \(w_m\), which converges weakly in \(H^1\) to \(w\in H^1, w\times n=0\) on \(\partial \varOmega \). The convergence holds in \(L^2\) strongly and then

$$\begin{aligned} \Vert w\Vert _{L^2}=1. \end{aligned}$$
(2.36)

On the other hand by (2.33),

$$\begin{aligned} \mathrm {div}\,w=0, \mathrm {rot}\,w=0. \end{aligned}$$
(2.37)

And therefore

$$\begin{aligned} w:=\mathrm {rot}\,v. \end{aligned}$$
(2.38)
$$\begin{aligned} 0=\int \limits v\mathrm {rot}\,w\mathrm {d}x=\int \limits w\mathrm {rot}\,v\mathrm {d}x-\int \limits v\cdot (w\times n)\mathrm {d}S=\int \limits w^2\mathrm {d}x, \end{aligned}$$

which gives

$$\begin{aligned} w=0\ \ \text{ in }\ \ \varOmega . \end{aligned}$$
(2.39)

This gives a contradiction with (2.36). \(\square \)

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Fan, J., Zhang, Z. & Zhou, Y. Local well-posedness for the incompressible full magneto-micropolar system with vacuum. Z. Angew. Math. Phys. 71, 42 (2020). https://doi.org/10.1007/s00033-020-1267-z

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Keywords

  • Well-posedness
  • Magneto-micropolar
  • Vacuum

Mathematics Subject Classification

  • 35A07
  • 35Q35
  • 76N15