Abstract
We investigate compressible micropolar fluids on a time-dependent domain with slip boundary conditions. Our contribution in this paper is threefold. Firstly, we establish the local existence of the strong solution. Secondly, the global existence of weak solutions is shown. The third one is the weak-strong uniqueness principle for slip boundary conditions. There are several new ideas developed by us to overcome the difficulties caused by the coupled terms and slip boundary conditions.
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Notes
Originally, the method was introduced for fixed domain, see [34].
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Acknowledgements
The work of B.-K. Huang is supported by the grant from NSFC under contract 11901148 and “the Fundamental Research Funds for the Central Universities.” Š. Nečasová has been supported by the Czech Science Foundation (GAČR) project GA19-04243 S. The Institute of Mathematics, CAS, is supported by RVO:67985840. The work of L. Zhang is supported by NSFC under contract 12101472 and “the Fundamental Research Funds for the Central Universities (WUT:2021IVA062).” The authors gratefully acknowledge the anonymous referees for their valuable comments.
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Appendix
Appendix
Proof of Lemma 4.2and Lemma 5.2. We observe that the stress tension for the velocity is coupled with micro-rotational velocity; however, the stress tension of angular-rotational velocity is decoupled. Hence, we should first construct the micro-rotational velocity; then, we use it to construct the velocity of fluids. We look for the extension of the boundary data satisfying the following conditions:
For \(\varvec{w}^b\), it should hold that
and
For \(\varvec{u}^b\), it should hold that
and
Firstly, we flatten the boundary and choose a proper smooth cutoff function which enjoy the regularity of the boundary. The detail can be found in [25]. Therefore, we can choose
Then we construct \({\tilde{\varvec{u}}}^b\) and \({\tilde{\varvec{w}}}^b\) satisfying (10.3) and (10.1), respectively. Moreover, in the following step we use it to define \(\varvec{w}^b\). Lastly, with \(\varvec{w}^b\) in hand, we define \(\varvec{u}^b\).
The construction of \({\tilde{\varvec{u}}}^b\) is directly borrowed from Appendix A in [25]. We list the result as follows
Now, we consider \({\tilde{\varvec{w}}}^b\). Precisely, for any \((y_1,y_2,y_3)\in \Omega _0\), we define
Moreover, the normal component of \({\tilde{\varvec{w}}}^{b}\) can be defined as
For the first term of right-hand side in (10.6), we have
Thanks to (1.6), we get
We only estimate the first term of right-hand side in (10.7); the rest terms can be done in the same way,
Finally, we get
For the second term of right-hand side in (10.6), we denote \(E\varvec{Y}=\nabla _x\varvec{Y}-{\mathbb {I}}\); then, it holds that
It is easy to observe that
where E is continuous, \(E(0)=0\). Hence, we have
Combining (10.9) and (10.11), we obtain
The tangential parts have similar bounds.
Now we construct \(\varvec{w}^b\) and rewrite (10.1) in a compact form
where \(E^1\) and \(E^2\) are sufficiently regular matrix and vector functions. Meanwhile, we rewrite (10.2) as
First, we take \(w_3^b={\tilde{w}}_3^b\). Next we construct \(w_1^b\), since \(w_2^b\) can be constructed in the same way. In particular,
Substituting the above identity into (10.14), we have
Then we divide \(w_1^b\) into two parts such that \(w_1^b=w_1^{b1}+w_1^{b2}\) where
here
Applying \(\partial _{y_3}\) to (10.17), we get
So, subtracting (10.19) from (10.16) gives that
Hence, we can define \(w_{1}^{b2}\) as follows
Finally, \(w_{1}^b=w_{1}^{b1}+w_{1}^{b2}\) is constructed as required relations. With \(\varvec{w}^b\) in hand, we can construct \(\varvec{u}^b\) in the same way which depends on \(\varvec{w}^b\). We also rewrite (10.3) in a compact way
where \(E^3\) and \(E^4\) are regular matrix and vector functions. And, (10.4) gives that
Also, we take \(u_3^b={{\tilde{u}}}_3^b\), then
We define \(u_1^b=u_{1}^{b1}+u_{1}^{b2}\) where
with
Hence, we have
Finally, we can define \(u_{1}^{b2}\) as follows
We notice that \(u_1^{b}=u_1^{b1}+u_1^{b2}\) satisfies (10.3) and (10.4).
Note that \(B^1_{ij}\), \(B^2_{ij}\), \(C^1_{ij}\) and \(C^2_{ij}\) are made of \(\tau (y)-\tau (\varvec{X}(y))\), \({\varvec{n}}(y)-{\varvec{n}}(\varvec{X}(y))\) and \(\nabla _x\varvec{Y}-{\mathbb {I}}\). Thanks to (10.8) and (10.10), we obtain the estimate for \(w_1^{b1}\) and \(u_1^{b1}\). When estimates come to \(w_1^{b2}\) and \(u_1^{b2}\), we use the fact
Therefore, repeating all the argument as before, we can derive the bound for \(\Vert \varvec{u}_{tt}^b\Vert _{L^2(0,T;L^2(\Omega _0))}\) and \(\Vert \varvec{w}_{tt}^b\Vert _{L^2(0,T;L^2(\Omega _0))}\). Other components in \({\mathcal {Y}}(T)\) are obtained in a similar way; we omit the details.
Lemma 4.1
[11] (Fundamental lemma) Let \(\rho \in L^\infty (0,T;L^2(B))\), \(\rho \ge 0\), \(\varvec{u}\in L^2(0,T;W_{0}^{1,2}(B;{\mathbb {R}}^3))\) be a weak solution of the continuity equation,
satisfying
for any \(\tau \in [0,T]\) and any test function \(\phi \in C_c^1([0,T]\times {\mathbb {R}}^3)\).
In addition, assume that \((\varvec{u}-\varvec{V})(\tau , \cdot )\cdot \varvec{n}|_{\Gamma _{\tau }}=0\) for \(\text{ a.a. } \tau \in (0, T),\) and that \(\rho _0\in L^2({\mathbb {R}}^3)\), \(\rho _0\ge 0\), \(\rho _0|_{B\backslash \Omega _0}=0\). Then
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Huang, B., Nečasová, Š. & Zhang, L. On the compressible micropolar fluids in a time-dependent domain. Annali di Matematica 201, 2733–2795 (2022). https://doi.org/10.1007/s10231-022-01218-6
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DOI: https://doi.org/10.1007/s10231-022-01218-6
Keywords
- Compressible micropolar fluids
- Asymmetric stress tensor
- Slip boundary conditions
- Strong solution
- Weak solutions