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

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.

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). $\square$